and singular $\mathcal{R}_s$ regions. Moreover, the structure of the singular region $\mathcal{R}_s$ of the SPEGBC yields the existence/non-existence of robust capture controls for the engagement (5.7),(2.52) and for the original engagement (2.51),(2.52). Furthermore, if these controls exist, the structure of the singular region of the SPEGBC also yields the structures of the corresponding capture zones.

## 5.3 Robust Capture Controls and Capture Zones

In this section, based on the SPEGBC space decomposition, the existence of the robust capture controls for the engagements (5.7),(2.52) and (2.51),(2.52) is established and the structure of the corresponding capture zones is derived.

**Remark 29.** Due to the results of Subsection 3.5.3, the SPEGBC space decomposition is based on the behaviour of the determining function. Using the equations (3.66) and (5.9), we obtain the determining function in the SPEGBC

$$ R(t) = R(t, \phi_{PC}, \phi_{E0}) \\ = \frac{a_{E,c}^{\max}(t)}{\tau_E} |\cos(\phi_{E0})| g_E(t) - \frac{a_{P,c}^{\max}(t)}{\tau_P} |\cos(\phi_{PC})| g_P(t), \quad t \in [t_0, t_f], \quad (5.11) $$

where

$$ g_l(t) = \int_{t_f}^{t} \left[ \exp\left(\frac{t-\xi}{\tau_l}\right) \frac{1}{V_l(\xi)} \int_{t_f}^{\xi} V_l(\sigma) d\sigma \right] d\xi, \quad t \in [t_0, t_f], \quad l = P, E. \quad (5.12) $$

Like in Subsection 3.5.3, we assume that the determining function $R(t) = R(t, \phi_{PC}, \phi_{E0})$ has a finite number of distinct zeros in the interval $(t_0, t_f)$.

**Remark 30.** Remember that the definitions of the robust capture control and its capture zone for the engagement (2.51),(2.52) are given in Section 2.6, while such notions for the engagement (5.7),(2.52) are not yet presented. However, the system (5.7) can be considered as a scalar particular case of the system (2.51) with zero coefficient for the scalar state variable, the scalar coefficients $h_u(t, \phi_{PC})$, $h_v(t, \phi_{E0})$ for the controls and zero non-homogeneity function. Therefore, the notions of the robust capture control and its capture zone for the engagement (5.7),(2.52) are quite similar to such notions for the engagement (2.51),(2.52). Definitions 2.6.2 - 2.6.4, valid for the engagement (2.51),(2.52), are directly reformulated for the engagement (5.7),(2.52). The existence of a robust capture control means the fulfilment of the conditions of Definition 2.6.3 for the engagement (2.51),(2.52) or the fulfilment of similar conditions in the case of the engagement (5.7),(2.52).
## 5.3.1 Main Assertions

The following assertion presents a criterion of the existence of the robust capture control for the engagement (5.7),(2.52).

**Theorem 5.3.1.** Robust capture control for the engagement (5.7),(2.52) exists if and only if the point $(t, z) = (t_f, 0)$ belongs to the closure of the singular region $\mathcal{R}_s$ of the SPEGBC, i.e.,

$$ (t_f, 0) \in \mathrm{clo}(\mathcal{R}_s). \qquad (5.13) $$

Proof of the theorem is presented in Subsection 5.5.1.

**Corollary 5.3.2.** Robust capture control for the engagement (5.7),(2.52) exists if and only if there exists a left-hand vicinity $(\hat{t}, t_f)$, $(t_0 \le \hat{t} < t_f)$ of the point $t = t_f$ such that the following inequality is valid:

$$ R(t) < 0 \quad \forall t \in (\hat{t}, t_f). \qquad (5.14) $$

*Proof.* Due to the inequality (5.14) and the decomposition of the SPEGBC space (see Subsection 3.5.3), the positive and negative boundaries of the regular region of the SPEGBC contain the point $(t, z) = (t_f, 0)$. Therefore, the inclusion (5.13) is valid in the SPEGBC, meaning that the corollary is a direct consequence of Theorem 5.3.1. □

The following assertion presents a sufficient condition for the validity of the inequality (5.14).

**Lemma 5.3.3.** Let the functions $V_P(t)$ and $V_E(t)$ be continuously differentiable, while the functions $a_{P,c}^{\max}(t)$ and $a_{E,c}^{\max}(t)$ be twice continuously differentiable in some left-hand vicinity of $t = t_f$ including this point. Let the following condition be satisfied:

$$ \frac{|\cos(\phi_{E0})|}{\tau_E} a_{E,c}^{\max}(t_f) - \frac{|\cos(\phi_{PC})|}{\tau_P} a_{P,c}^{\max}(t_f) < 0. \qquad (5.15) $$

Then, the inequality (5.14) is valid and, therefore, a robust capture control for the engagement (5.7),(2.52) exists.

Proof of the lemma is presented in Subsection 5.5.2.

Theorem 5.3.1, Corollary 5.3.2 and Lemma 5.3.3 establish the conditions of the existence of a robust capture control for the engagement (5.7),(2.52). Now, we are going to design two types of such controls and to construct the corresponding capture zones subject to the validity of the condition (5.14).
It is clear that $\int_{t_f}^t R(\sigma)d\sigma \ge 0$ for all $t \in [\hat{t}, t_f]$. However, $\hat{t} \in [t_0, t_f)$ is not, in general, the minimal possible such time instant.

By $t_{\min}$, let us denote the minimal value of $t \in [t_0, t_f)$ such that the following inequality is satisfied:

$$z_c(t) \triangleq \int_{t_f}^t R(\sigma)d\sigma \ge 0 \quad \forall t \in [t_{\min}, t_f]. \qquad (5.16)$$

**Remark 31.** If $t_{\min} > t_0$, then the function $z_c(t)$ changes its sign at $t = t_{\min}$ and $z_c(t) < 0$ in some left-hand vicinity of this point. Moreover, $R(t) > 0$ in some neighbourhood of $t = t_{\min}$. Since the function $R(t)$ has a finite number of distinct zeros in the interval $(t_0, t_f)$, the function $z_c(t)$ has a finite number of distinct zeros in the interval $(t_{\min}, t_f)$. Let there be $l \ge 0$ such zeros $t_{c,1} < \dots < t_{c,l}$.

In the plane $(t, z)$, let us consider the set

$$C_z \triangleq \{(t, z) : t \in [t_{\min}, t_f], |z| \le z_c(t)\}. \qquad (5.17)$$

Also, we consider the control $u_{\text{sg}}^0(t, z)$ having the form (3.73) with $h_u(t) = h_u(t, \phi_{P0})$ and $0 < |\alpha_u(t)| \le 1$.

**Theorem 5.3.4.** Let the condition (5.14) be valid. Then, $u_{\text{sg}}^0(t, z)$ is a robust capture control from any point $(\bar{t}, \bar{z}) \in C_z$ for the engagement (5.7), (2.52). Moreover, the set $C_z$ is the capture zone of $u_{\text{sg}}^0(t, z)$, i.e., $C_z = C_z(u_{\text{sg}}^0(t, z))$.

Proof of the theorem is presented in Subsection 5.5.3.

Control $u_{\text{sg}}^0(t, z)$, considered in Theorem 5.3.4, is a bang-bang control with respect to $z$ for any $t \in [t_0, t_f)$, i.e., it is a discontinuous function with respect to $z$ for all $t \in [t_0, t_f)$. Now, we are going to propose a continuous with respect to $z$ robust capture control $u_{\text{sg,c}}(t, z)$ subject to the validity of the condition (5.14). Due to this condition, the time instant $t_{\min}$ and the set $C_z$, defined by the equations (5.16) and (5.17), respectively, exist. Construction of the aforementioned control $u_{\text{sg,c}}(t, z)$ is based on a function $L(t, z)$, $(t, z) \in [t_0, t_f] \times E^1$ satisfying the following conditions:

**AL1.** For any $t \in [t_0, t_f]$, the function $L(t, z)$ is odd with respect to $z \in E^1$.

**AL2.** For any $t \in [t_0, t_f]$, the function $L(t, z)$ and its partial derivative $\partial L(t, z)/\partial z$ are continuous with respect to $z \in E^1$.

**AL3.** For any $t \in [t_0, t_f]$, $\partial L(t, z)/\partial z \ge 0$.

**AL4.** For any $t \in (t_{\min}, t_f) : t \ne t_{c,i} (i = 1, \dots, l)$, $L(t, z_c(t)) \ge 1$.

Consider the pursuer's control

$$u_{\text{sg,c}}(t, z) = -\text{sign}(h_u(t, \phi_{P0}))\text{sat}(L(t, z)), \quad (t, z) \in [t_0, t_f] \times E^1. \qquad (5.18)$$
**Theorem 5.3.5.** Let the condition (5.14) and the conditions AL1 - AL4 be valid. Then, the control (5.18) is a robust capture control from any point $(\bar{t}, \bar{z}) \in C_z$ and the set $C_z$ is the capture zone of this control for the engagement (5.7), (2.52).

Proof of the theorem is presented in Subsection 5.5.4.

**Example 5.3.6.** In this example, we present the simplest version of the function $L(t, z)$. For this purpose, we consider the following three possible cases of the time instant $t_{\min}$: (i) $t_{\min} > t_0$; (ii) $t_{\min} = t_0$ and $z_c(t_0) = 0$; (iii) $t_{\min} = t_0$ and $z_c(t_0) > 0$.

Case (i). In this case, the simplest version of the function $L(t, z)$ is

$$L(t, z) = \begin{cases} \frac{z}{z_c(t)}, & t \in (t_{\min}, t_f) : t \neq t_{c,i} \ (i = 1, \dots, l), \quad z \in E^1, \\ z, & t \in [t_0, t_{\min}] \cup (\cup_{j=1}^l \{t = t_{c,j}\}) \cup \{t = t_f\}, \quad z \in E^1. \end{cases} \quad (5.19)$$

Case (ii). In this case, the simplest version of the function $L(t, z)$ is

$$L(t, z) = \begin{cases} \frac{z}{z_c(t)}, & t \in (t_0, t_f) : t \neq t_{c,i} \ (i = 1, \dots, l), \quad z \in E^1, \\ z, & t \in \{t = t_0\} \cup (\cup_{j=1}^l \{t = t_{c,j}\}) \cup \{t = t_f\}, \quad z \in E^1. \end{cases} \quad (5.20)$$

Case (iii). In this case, the simplest version of the function $L(t, z)$ is

$$L(t, z) = \begin{cases} \frac{z}{z_c(t)}, & t \in [t_0, t_f) : t \neq t_{c,i} \ (i = 1, \dots, l), \quad z \in E^1, \\ z, & t \in (\cup_{j=1}^l \{t = t_{c,j}\}) \cup \{t = t_f\}, \quad z \in E^1. \end{cases} \quad (5.21)$$

**Remark 32.** If $t_{\min}$, defined in the equation (5.16), coincides with $t_0$, then the set $C_z$ is the t-complete capture zone of the controls $u_{sg}^0(t, z)$ and $u_{sg,c}(t, z)$ mentioned in Theorems 5.3.4 and 5.3.5. For example, if the condition (5.14) is valid and the determining function $R(t)$ changes its sign once in the interval $(t_0, t_f)$, then the fulfilment of the inequality $\int_{t_f}^{t_0} R(\sigma)d\sigma \ge 0$ is necessary and sufficient for the set $C_z$ to be the t-complete capture zone of the controls $u_{sg}^0(t, z)$ and $u_{sg,c}(t, z)$. Another (simpler) example is the validity of the condition (5.14) with $\hat{t} = t_0$, which guarantees the t-completeness for the capture zone $C_z$ of these controls.

Theorem 5.3.1, Corollary 5.3.2 and Lemma 5.3.3 establish various necessary and sufficient conditions for the existence of a robust capture control for the engagement (5.7), (2.52). Theorems 5.3.4 and 5.3.5 present two robust capture controls and their capture zones for this engagement. Now, we are going to present several assertions with respect to robust capture controls and their capture zones for the engagement (2.51), (2.52).
**Theorem 5.3.7.** Let $u_{\text{sg}}(t, z)$ be a robust capture control from the point $(\bar{t}, \bar{z}) \in [t_0, t_f) \times E^1$ for the engagement (5.7), (2.52). Then $u(t, x) = u_{\text{sg}}(t, z(t, x))$, where $z(t, x)$ is given by (5.6), is a robust capture control for the engagement (2.51), (2.52) from any point $(\bar{t}, \bar{x}) \in [t_0, t_f) \times E^6$ satisfying the condition $z(\bar{t}, \bar{x}) = \bar{z}$.

*Proof.* The statement of the theorem directly follows from Definition 2.6.3, Remark 30 and the connection (5.6) between the scalar state variable $z$ of the engagement (5.7), (2.52) and the 6-dimensional vector-valued state variable $x$ of the engagement (2.51), (2.52). $\square$

Based on the controls $u_{\text{sg}}^0(t, z)$ and $u_{\text{sg,c}}(t, z)$ mentioned in Theorems 5.3.4 and 5.3.5, we construct the following controls for the engagement (2.51), (2.52):

$$u^0(t, x) = u_{\text{sg}}^0(t, z(t, x)), \quad (t, x) \in [t_0, t_f] \times E^6, \qquad (5.22)$$

$$u_c(t, x) = u_{\text{sg,c}}(t, z(t, x)), \quad (t, x) \in [t_0, t_f] \times E^6, \qquad (5.23)$$

where the function $z(t, x)$ is given by the equation (5.6). Also, based on the set $C_z \subset S_{[t_0, t_f]}$, we consider the set

$$C_x \triangleq \{(t, x) \in [t_0, t_f] \times E^6 : (t, z(t, x)) \in C_z\}. \qquad (5.24)$$

The following assertions are direct consequences of Theorems 5.3.4, 5.3.5, 5.3.7, Remark 32 and the connection (5.6) between the state variable $z$ of the engagement (5.7), (2.52) and the state variable $x$ of the engagement (2.51), (2.52).

**Corollary 5.3.8.** Let the condition (5.14) be valid. Then, $u^0(t, x)$, given by (5.22), is a robust capture control from any point $(\bar{t}, \bar{x}) \in C_x$ for the engagement (2.51), (2.52). Moreover, the set $C_x$ is the capture zone of $u^0(t, x)$, i.e., $C_x = C_x(u^0(t, x))$. This capture zone is t-complete if the capture zone $C_z(u_{\text{sg}}^0(t, z))$ mentioned in Theorem 5.3.4 is t-complete.

**Corollary 5.3.9.** Let the condition (5.14) and the conditions AL1 - AL4 be valid. Then, the control $u_c(t, x)$, given by (5.23), is a robust capture control from any point $(\bar{t}, \bar{x}) \in C_x$ and the set $C_x$ is the capture zone of this control for the engagement (2.51), (2.52), i.e., $C_x = C_x(u_c(t, x))$. This capture zone is t-complete if the capture zone $C_z(u_{\text{sg,c}}(t, z))$ mentioned in Theorem 5.3.5 is t-complete.
### 5.3.2 Construction of the Function $L(t, z)$ Based on Linear-Quadratic Differential Game with Cheap Controls

In this section, we will construct another (than in Example 5.3.6) function $L(t, z)$ satisfying the conditions AL1-AL4.

**Remark 33.** The aforementioned construction is carried out subject to the assumption that the function $z_c(t)$, defined in the equation (5.16), does not have zeros in the interval $(t_{\min}, t_f)$, i.e., the number $l$ (see Remark 31) equals zero and $z_c(t) > 0$, $t \in (t_{\min}, t_f)$. This assumption is reasonable from the practical viewpoint and it is valid in many cases of the pursuit-evasion engagement between two flying vehicles (see e.g. [4, 5, 10, 13] and references therein). The simple sufficient condition for the validity of this assumption is that the determining function $R(t)$ has no more than one zero in the interval $(t_{\min}, t_f)$. The equivalent sufficient condition is that the derivative $dR(t)/dt$ is negative in the interval $(t_{\min}, t_f)$.

Proceed to the construction of the function $L(t, z)$. First, we construct an auxiliary function. The latter construction, consisting of three stages, is based on the analysis and solution of the scalar zero-sum linear-quadratic differential game with cheap controls (see Chapter 4). At the first stage, we use the minimizer's component of the saddle-point solution to this game (see the equation (4.15)) and replace there $h_u(t)$ and $h_v(t)$ with the functions $h_u(t, \phi_{PC})$ and $h_v(t, \phi_{E0})$ (see the equation (5.9)). Moreover, we replace the time instant $t_0$ with the time instant $t_{\min}$ defined in the equation (5.16). At the second stage, we multiply the resulting expression by $-\text{sign}(h_u(t, \phi_{PC}))$. Thus, we obtain the following function for all $(t, z) \in [t_{\min}, t_f] \times E^1$:

$$ \tilde{L}(t, z, \lambda, \varepsilon) = \frac{|h_u(t, \phi_{PC})|}{\varepsilon\lambda + \int_t^{t_f} \left[ (h_u(\sigma, \phi_{PC}))^2 - \lambda(h_v(\sigma, \phi_{E0}))^2 \right] d\sigma} z, \quad (5.25) $$

where $\lambda$ is some given positive number; $\varepsilon > 0$ is a small parameter.

Let us check the feasibility of the function $\tilde{L}(t, z, \lambda, \varepsilon)$ for all arbitrarily small values $\varepsilon > 0$. For this purpose, we introduce the function

$$ H_{uv}(t, \phi_{PC}, \phi_{E0}) \triangleq \frac{\int_t^{t_f} (h_u(\sigma, \phi_{PC}))^2 d\sigma}{\int_t^{t_f} (h_v(\sigma, \phi_{E0}))^2 d\sigma}, \quad t \in [t_{\min}, t_f]. \quad (5.26) $$

Due to the results of Section 4.4 (see the equation (4.12) and Lemmas 4.4.2, 4.4.3), the function $\tilde{L}(t, z, \lambda, \varepsilon)$ is feasible for all arbitrarily small values $\varepsilon > 0$ if the following inequality is valid:

$$ \lambda_{\inf}(\phi_{PC}, \phi_{E0}) \triangleq \inf_{t \in [t_{\min}, t_f]} H_{uv}(t, \phi_{PC}, \phi_{E0}) > 0, \quad (5.27) $$
and the positive number $\lambda$ satisfies the inequality

$$ \lambda \leq \lambda_{\inf}(\phi_{PC}, \phi_{E0}). \qquad (5.28) $$

The following assertions establish the validity of the inequality (5.27).

**Lemma 5.3.10.** Let the functions $V_P(t)$ and $V_E(t)$ be continuously differentiable in some left-hand vicinity of $t = t_f$ including this point. Let $\cos(\phi_{PC}) \neq 0$ and $\cos(\phi_{E0}) \neq 0$. Then, the inequality (5.27) is valid. Moreover, if the positive number $\lambda$ satisfies the strict version of the inequality (5.28), the integral in the denominator of the expression (5.25) for $\tilde{L}(t, z, \lambda, \varepsilon)$ is positive for all $t \in [t_{\min}, t_f)$.

Proof of the lemma is presented in Subsection 5.5.5.

Quite similarly to Corollary 4.4.4 (see Section 4.4), we have the assertion.

**Corollary 5.3.11.** Let $\cos(\phi_{PC}) \neq 0$ and $\cos(\phi_{E0}) \neq 0$. Let $H_{uv}(t, \phi_{PC}, \phi_{E0})$ be an increasing function in the interval $[t_{\min}, t_f)$. Then, $\lambda_{\inf}(\phi_{PC}, \phi_{E0}) = H_{uv}(t_{\min}, \phi_{PC}, \phi_{E0}) \triangleq \lambda_{\min} > 0$. If $\lambda = \lambda_{\min}$, the integral in the denominator of the expression (5.25) for $\tilde{L}(t, z, \lambda, \varepsilon)$ is positive for all $t \in (t_{\min}, t_f)$, while for $t = t_{\min}$ and $t = t_f$, this integral becomes zero.

**Proposition 5.3.12.** Let $\cos(\phi_{PC}) \neq 0$ and $\cos(\phi_{E0}) \neq 0$. Let the function

$$ \tilde{h}_{uv}(t, \phi_{PC}, \phi_{E0}) \triangleq \frac{|h_u(t, \phi_{PC})|}{|h_v(t, \phi_{E0})|} \qquad (5.29) $$

be increasing for $t \in [t_{\min}, t_f)$. Then $H_{uv}(t, \phi_{PC}, \phi_{E0})$ is an increasing function in the interval $[t_{\min}, t_f)$.

Proof of the proposition is presented in Subsection 5.5.6.

Proceed to the third stage in the construction of the aforementioned auxiliary function. At this stage, we calculate the limit for $\varepsilon \to +0$ of the function $\tilde{L}(t, z, \lambda, \varepsilon)$ given by (5.25). This limit has the form

$$ L_{\text{aux}}(t, z, \lambda) \triangleq \tilde{L}(t, z, \lambda, 0) = \frac{|h_u(t, \phi_{PC})|}{\int_t^{t_f} [(h_u(\sigma, \phi_{PC}))^2 - \lambda(h_v(\sigma, \phi_{E0}))^2] d\sigma} z. \qquad (5.30) $$

**Remark 34.** By virtue of Lemma 5.3.10, if the positive number $\lambda$ satisfies the strict version of the inequality (5.28), the t-dependent gain for $z$ in the function $L_{\text{aux}}(t, z, \lambda)$ is positive for all $t \in [t_{\min}, t_f)$. For $t \to t_f - 0$, this gain tends to $+\infty$. Moreover, by virtue of Corollary 5.3.11, if $H_{uv}(t, \phi_{PC}, \phi_{E0})$ is an increasing function in the interval $[t_{\min}, t_f)$ and $\lambda = \lambda_{\min}$, then the t-dependent gain for $z$ in the function $L_{\text{aux}}(t, z, \lambda)$ is positive for all $t \in (t_{\min}, t_f)$. For $t \to t_{\min} + 0$ and for $t \to t_f - 0$, this gain tends to $+\infty$.
Now, using the auxiliary function $L_{aux}(t, z, \lambda)$, we are going to construct the function $L(t, z)$ satisfying the condition AL1 - AL4. For this purpose, like in Example 5.3.6, we distinguish three possible cases of the time instant $t_{min}$. Namely, (a) $t_{min} > t_0$; (b) $t_{min} = t_0$ and $z_c(t_0) = 0$; (c) $t_{min} = t_0$ and $z_c(t_0) > 0$. In all these cases, we assume that $\cos(\phi_{PC}) \neq 0$ and $\cos(\phi_{E0}) \neq 0$.

**Case (a):** $t_{min} > t_0$

In this case, we assume that the function $\tilde{h}_{uv}(t, \phi_{PC}, \phi_{E0})$ increases in the interval $[t_{min}, t_f)$. Also, we introduce into the consideration the following numbers:

$$ \tilde{\gamma}_a = \inf_{t \in (t_{min}, t_f)} L_{aux}(t, z_c(t), \lambda_{min}) > 0, \quad \gamma_a = \max \left\{ 1, \frac{1}{\tilde{\gamma}_a} \right\}. \quad (5.31) $$

Based on these assumption and numbers, the function $L(t, z)$ is represented as:

$$ L(t, z) = \begin{cases} \gamma_a L_{aux}(t, z, \lambda_{min}), & t \in (t_{min}, t_f), \quad z \in E^1, \\ z, & t \in [t_0, t_{min}] \cup \{t = t_f\}, \quad z \in E^1. \end{cases} \quad (5.32) $$

**Case (b):** $t_{min} = t_0$ and $z_c(t_0) = 0$, $R(t_0) \neq 0$

In this case, we assume that the function $\tilde{h}_{uv}(t, \phi_{PC}, \phi_{E0})$ increases in the interval $[t_0, t_f)$. Also, we introduce into the consideration the following numbers:

$$ \lambda_0 = H_{uv}(t_0, \phi_{P0}, \phi_{E0}), \\ \tilde{\gamma}_b = \inf_{t \in (t_0, t_f)} L_{aux}(t, z_c(t), \lambda_0) > 0, \quad \gamma_b = \max \left\{ 1, \frac{1}{\tilde{\gamma}_b} \right\}. \quad (5.33) $$

Using these assumption and numbers, we construct the function $L(t, z)$ in the form

$$ L(t, z) = \begin{cases} \gamma_b L_{aux}(t, z, \lambda_0), & t \in (t_0, t_f), \quad z \in E^1, \\ z, & t \in \{t = t_0\} \cup \{t = t_f\}, \quad z \in E^1. \end{cases} \quad (5.34) $$

**Case (c):** $t_{min} = t_0$ and $z_c(t_0) > 0$

In this case, we introduce into the consideration the following numbers:

$$ \tilde{\gamma}_c(\lambda) = \inf_{t \in [t_0, t_f)} L_{aux}(t, z_c(t), \lambda) > 0, \quad \gamma_c(\lambda) = \max \left\{ 1, \frac{1}{\tilde{\gamma}_c(\lambda)} \right\}, \quad (5.35) $$
where $0 < \lambda < \lambda_{inf}(\phi_{PC}, \phi_{E0})$ is any prechosen number.

Using these numbers, we construct the function $L(t, z)$ as follows:

$$L(t, z) = \begin{cases} \gamma_c(\lambda)L_{aux}(t, z, \lambda), & t \in [t_0, t_f), \quad z \in E^1, \\ z, & t = t_f, \quad z \in E^1. \end{cases} \quad (5.36)$$

## 5.4 Numerical Examples: Pursuit-Evasion Engagement of Two Flying Vehicles

### 5.4.1 Example 1: Two Changes of Sign of the Determining Function

**Data of the engagement and the behaviour of the determining function**

Consider the engagements (5.7),(2.52) and (2.51),(2.52) with the following set of their data:

$$V_P(t) = 2000 \exp \left( \frac{t}{16} \right) \text{ m/sec}, \quad V_E(t) \equiv 2600 \text{ m/sec}, \quad t_0 = 0,$$

$$\phi_{E0} = \frac{5\pi}{6}, \quad R_0 = 30000 \text{ m}, \quad \tau_P = 0.1 \text{ sec}, \quad \tau_E = 0.15 \text{ sec},$$

$$a_{P,c}^{\max}(t) = (25t + 100) \text{ m/sec}^2, \quad a_{E,c}^{\max}(t) = (125 \arctan(t) + 50) \text{ m/sec}^2. \quad (5.37)$$

For these data, the equation (2.7) has the unique solution $t_C = t_{C,min} = 6.845579 \text{ sec}$ which, along with (2.8),(2.9),(2.34), yields

$$\cos(\phi_{PC}) = 0.853688, \quad \sin(\phi_{PC}) = 0.520785, \quad t_f = 6.845579 \text{ sec}. \quad (5.38)$$

Based on the data (5.37),(5.38) and using the equations (5.11),(5.12), we obtain the determining function in this example

$$R(t) = \frac{\sqrt{3}(125 \arctan(t) + 50)}{0.3} g_E(t) - 8.53688(25t + 100)g_P(t),$$

$$g_E(t) = 0.15 \left[ 6.695579 - t + 0.15 \exp \left( \frac{t - 6.845579}{0.15} \right) \right],$$

$$g_P(t) = 0.009938 \exp(10(t - 6.845579)) + 1.590062 \exp\left(\frac{6.845579 - t}{16}\right) - 1.6,$$

where $t \in [0, 6.845579]$.

Graph of this determining function is presented in Fig. 5.1. It is seen that $R(t)$ has two distinct zeros ($t_{s1} = 1.03811$ and $t_{s2} = 3.76126$) in the interval $(0, 6.845579)$, and $R(t)$ changes its sign twice in this interval. Moreover, the function $R(t)$ is negative in the interval $(t_{s2}, t_f) = (3.76126, 6.845579)$.
<figure><img src="image_27.png" /><figcaption>Figure 5.1: Graph of the determining function $R(t)$ in Example 1 (Subsection 5.4.1)</figcaption></figure>

Decomposition of the pursuit-evasion game's space

Using the results of Subsection 3.5.3, we are going to decompose the space of the SPEGBC.

<figure><img src="image_28.png" /><figcaption>Figure 5.2: Graph of $\int_{t_f}^t R(\sigma)d\sigma$ in Example 1 (Subsection 5.4.1)</figcaption></figure>

Due to the aforementioned behaviour of $R(t)$, this decomposition should be done in accordance with Subcase 2.2. However, for a more precise analysis, we should check whether the following equation:

$$ \int_{t_f}^{t} R(\sigma)d\sigma = 0 \qquad (5.39) $$
has the unique solution $t = t_{in}$ in the interval $(t_{s1}, t_{s2})$. Graph of the function $\int_{t_f}^{t} R(\sigma)d\sigma$, $t \in [0, t_f]$, depicted in Fig. 5.2, directly shows the existence of such a solution, and $t_{in} = 1.883973$ (the additional root of (5.39) is $\bar{t} = 0.504777 \in (0, t_{s1})$). Thus, to decompose the space of the SPEGBC, we should use the results of Subcase 2.2.1 (see the equations (3.92)-(3.95) and Fig. 3.13). This yields Fig. 5.3.

<figure><img src="image_29.png" /><figcaption>Figure 5.3: Decomposition of the space of the SPEGBC in Example 1 (Subsection 5.4.1)</figcaption></figure>

## Construction of the robust capture controls and capture zones

Figs. 5.3 and 5.1 directly show the fulfilment of the conditions of Theorem 5.3.1 and Corollary 5.3.2. Moreover, using the data (5.37),(5.38), we obtain that the value of the expression in the left-hand side of the inequality (5.15) is negative: -997.050078, meaning that in the present example this inequality is satisfied. Thus, by virtue of any of the assertions, either Theorem 5.3.1 or Corollary 5.3.2 or Lemma 5.3.3, a robust capture control for the engagement (5.7),(2.52) exists in this example. Now, let us construct the capture zones of the controls $u_{sg}^0(t, z)$ and $u_{sg,c}(t, z)$, mentioned in Theorems 5.3.4 and 5.3.5 for the engagement (5.7),(2.52). To do this construction, we should obtain the value $t_{min}$ and the function $z_c(t)$ defined by the equation (5.16). From Fig. 5.2, it is seen that the function $\int_{t_f}^{t} R(\sigma)d\sigma$ is nonnegative in the interval $[t_{in}, t_f]$, while in the interval $(\bar{t}, t_{in})$ this function is negative. Thus, $t_{min} = t_{in}$ and $z_c(t) = \int_{t_f}^{t} R(\sigma)d\sigma$, $t \in [t_{min}, t_f]$, $t_{min} = 1.883973$, $t_f = 6.845579$. Using the function $z_c(t)$, we construct the capture zone $C_z$ of the control $u_{sg}^0(t, z)$ mentioned in Theorem 5.3.4. This capture zone is presented in Fig 5.4.
<figure><img src="image_30.png" /><figcaption>Figure 5.4: Capture zone $C_z$ in Example 1 (Subsection 5.4.1)</figcaption></figure>

Using the expression of the function $h_u(t, \phi_{PC})$ (see the equation (5.9)) and the data (5.37),(5.38), we obtain that in this example $h_u(t, \phi_{PC}) < 0, t \in [0, t_f)$ and $h_u(t_f, \phi_{PC}) = 0$. Therefore, the aforementioned control $u_{sg}^0(t, z)$ has the form

$$ u_{sg}^0(t, z) = \begin{cases} \operatorname{sign}(z), & t \in [0, t_f), \ z \neq 0, \\ -\bar{h}_u \operatorname{sign}(z), & t = t_f, \ z \neq 0, \\ \alpha_u(t) \operatorname{sign}(h_u(t)), & t \in [0, t_f], \ z = 0, \end{cases} \quad (5.40) $$

where $t_f = 6.845579$, the value $\bar{h}_u$ is defined in Remark 10, $\alpha_u(t)$ is any function given in the interval $[0, t_f]$ and satisfying the inequality $0 < |\alpha_u(t)| \le 1$.

Note that, by virtue of Theorem 5.3.5, the set $C_z$ presented in Fig 5.4 is the capture zone not only of the control $u_{sg}^0(t, z)$, but also of the control $u_{sg,c}(t, z)$ (see the equation (5.18)) where the function $L(t, z)$ satisfies the conditions AL1 - AL4. In this example, the function $L(t, z)$ can be chosen due to the equation (5.19) as:

$$ L(t, z) = \begin{cases} \frac{z}{z_c(t)}, & t \in (t_{\min}, t_f), \quad z \in E^1, \\ z, & t \in [0, t_{\min}] \cup \{t = t_f\}, \quad z \in E^1, \end{cases} \quad (5.41) $$

where $t_{\min} = 1.883973, t_f = 6.845579$.

In this example, the function $\tilde{h}_{uv}(t, \phi_{PC}, \phi_{E0})$ increases for $t \in [t_{\min}, t_f] = [1.883973, 6.845579]$ (see Fig. 5.5) meaning that $\lambda_{\inf}(\phi_{PC}, \phi_{E0}) = \lambda_{\min}$ (see Corollary 5.3.11 and Proposition 5.3.12).

In Fig. 5.6, the graphs of the function $L_{aux}(t, z_c(t), \lambda_{\min})$ and the function $\gamma_a L_{aux}(t, z_c(t), \lambda_{\min})$ are depicted, where $L_{aux}(t, z, \lambda)$ is given by the equation
<figure><img src="image_31.png" /><figcaption>Figure 5.5: Graph of $\tilde{h}_{uv}(t, \phi_{PC}, \phi_{E0})$ in Example 1 (Subsection 5.4.1)</figcaption></figure>

(5.30); $h_P(t, \phi_{PC})$ and $h_E(t, \phi_{E0})$ are given by the equation (5.9) and the data (5.37)-(5.38); $\lambda_{min}$ is defined in Corollary 5.3.11 and in this example $\lambda_{min} = 0.953762$; $\gamma_a$ is defined in the equation (5.31) and in this example $\gamma_a = 1.4456$.

<figure><img src="image_32.png" /><figcaption>Figure 5.6: Graphs of $L_{aux}(t, z_c(t), \lambda_{min})$ and $\gamma_a L_{aux}(t, z_c(t), \lambda_{min})$ in Example 1 (Subsection 5.4.1)</figcaption></figure>

Therefore, the function $L(t, z)$ also can be chosen due to the equation (5.32) as:

$$L(t, z) = \begin{cases} \gamma_a L_{aux}(t, z, \lambda_{min}), & t \in (t_{min}, t_f), \quad z \in E^1, \\ z, & t \in [0, t_{min}] \cup \{t = t_f\}, \quad z \in E^1. \end{cases} \quad (5.42)$$
Based on the equation (5.18) and taking into account the positiveness of $\cos(\phi_{PC})$, we obtain the control $u_{sg,c}(t, z)$ in the form

$$u_{sg,c}(t, z) = \begin{cases} \text{sat}(L(t, z)), & t \in [0, t_f), \quad z \in E^1, \\ -\bar{h}_u \text{sat}(L(t, z)), & t = t_f, \quad z \in E^1. \end{cases} \quad (5.43)$$

In Fig. 5.7, the trajectories of the scalar pursuit-evasion engagement, governed by the differential equation in (5.7), are presented. All these trajectories start from the point $(t = 4, z = 20) \in C_z$. One of these trajectories (the solid line) is generated by the robust capture control $u(t) = u_{sg}^0(t, z)$ (see the equation (5.40) with $\alpha_u(t) \equiv 1$) and by the evader's control $v(t) = v_{sg}^0(t, z)$ (see the equation (3.74) with $h_v(t) = h_v(t, \phi_{E0})$ and $\alpha_v(t) \equiv 1$). The two other trajectories (the dotted and dashed lines) are generated by the robust capture control $u(t) = u_{sg,c}(t, z)$ (with the function $L(t, z)$ given in the equations (5.41) and (5.42)) and by the aforementioned evader's control. It is seen that, starting in the capture zone $C_z$, all these trajectories remain in this set till the end of the engagement. Thus, these trajectories reach the point $(t, z) = (t_f, 0) \in C_z$. It also is seen that the use of the robust capture control $u(t) = u_{sg}^0(t, z)$ yields the sliding mode of the trajectory along the axis $t$ for $t \in [5, t_f]$. The trajectory, generated by the robust capture control $u(t) = u_{sg,c}(t, z)$ ($L(t, z)$ is given in (5.41)), coincides with the positive boundary of $C_z$ for $t \in [4.642626, t_f]$. The trajectory, generated by the robust capture control $u(t) = u_{sg,c}(t, z)$ ($L(t, z)$ is given in (5.42)), remains strictly inside $C_z$ for $t \in [4, t_f]$.

In Fig. 5.8, the time realizations of the aforementioned robust capture controls along the corresponding trajectories of the scalar pursuit-evasion engagement are presented. It is seen that the control $u(t) = u_{sg}^0(t, z)$ yields the chattering of its time realization for $t \in [5, t_f]$, while the time realization of the control $u(t) = u_{sg,c}(t, z)$ is completely chattering free.

**Remark 35.** Let us note the following. Due to the equation (3.74) with $h_v(t) = h_v(t, \phi_{E0})$ and $\alpha_v(t) \equiv 1$, the equation (5.9), and the data (5.37) of this example, we directly obtain that the time realization of the aforementioned evader's control $v_{sg}^0(t, z)$ along each of the trajectories $z = z(t)$ presented in Fig. 5.7 is $v_{sg}^0(t, z(t)) \equiv -1$, $t \in [0, t_f]$. Therefore, all the results of Figs. 5.7 and 5.8, remain the same in the case where $v(t) = v_{sg}^0(t, z)$ is replaced with $v(t) \equiv -1$, while the latter is an admissible evader's control in accordance with Definition 2.6.1.

Proceed to the construction of robust capture controls and corresponding capture zones for the engagement (2.51),(2.52). Due to the equations (5.22),(5.23),(5.24) and Corollaries 5.3.8, 5.3.9, to do this construction we
<figure><img src="image_33.png" /><figcaption>Figure 5.7: Trajectories of scalar pursuit-evasion engagement in Example 1 (Subsection 5.4.1)</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Figure 5.8: Time realizations of the robust capture controls in Example 1 (Subsection 5.4.1)</figcaption></figure>

can use the controls (5.40),(5.43), the set $C_z$ (see Fig. 5.4) and the function $z(t, x)$ (see the equation (5.6)). Based on the data (5.37),(5.38), let us derive the function $z(t, x)$ in this example. For this purpose, we need to know the entries $\varphi_{1j}(t)$, ($j = 1, ..., 6$) of the vector-row $\varphi_1(t)$ and the vector-column $f(t, \phi_{PC}, \phi_{E0})$. Using the equations (5.4),(5.5), we obtain these entries in the form

$$ 
\begin{aligned}
\varphi_{11}(t) &= 1, & \varphi_{12}(t) &= 6.845579 - t, & t \in [0, 6.845579], \\
\varphi_{13}(t) &= -13.659008 \left[ \frac{1}{1610} \exp \left( \frac{t - 6.845579}{0.1} \right) \right. \\
&\qquad \left. + \frac{16}{161} \exp \left( \frac{6.845579 - t}{16} \right) - 0.1 \right], & t \in [0, 6.845579], \\
\varphi_{14}(t) &= 0.129904 \left[ t - 6.845579 + 0.15 \right]
\end{aligned}
$$
$$ -0.15 \exp\left(\frac{t - 6.845579}{0.15}\right)], \quad t \in [0, 6.845579], \\ \varphi_{15}(t) = 1707.376\left[(22.845484 - t) \exp\left(\frac{t}{16}\right) - 24.543118\right], \quad t \in [0, 6.845579], \\ \varphi_{16}(t) = 0, \quad t \in [0, 6.845579]. \tag{5.44} $$

For the data (5.37),(5.38), the vector-column $f(t, \phi_{PC}, \phi_{E0})$ (see the equations (2.47),(2.48)) becomes as:

$$ f(t, \phi_{PC}, \phi_{E0}) = \operatorname{col}(0, -65.098125 \exp(t/16), 0, 0, 0, 0), \quad t \in [0, 6.845579]. \tag{5.45} $$

Thus, the aforementioned function $z(t, x)$ has the form

$$ z(t, x) = x_1 + (6.845579 - t)x_2 + \varphi_{13}(t)x_3 + \varphi_{14}(t)x_4 + \varphi_{15}(t)x_5 \\ -1041.57[(t - 22.845484) \exp(t/16) + 24.543118], \quad t \in [0, 6.845579]. \tag{5.46} $$

Using the controls (5.40),(5.43) and the function (5.46), we construct the following robust capture controls for the engagement (2.51),(2.52):

$$ u^0(t, x) = \begin{cases} \operatorname{sign}(z(t, x)), & t \in [0, t_f], \quad x \in E^6, \quad z(t, x) \neq 0, \\ -\bar{h}_u \operatorname{sign}(z(t, x)), & t = t_f, \quad x \in E^6, \quad z(t, x) \neq 0, \\ \alpha_u(t) \operatorname{sign}(h_u(t)), & t \in [0, t_f], \quad x \in E^6, \quad z(t, x) = 0. \end{cases} \tag{5.47} $$

$$ u_c(t, x) = \begin{cases} \operatorname{sat}(L(t, z(t, x))), & t \in [0, t_f), \quad x \in E^6, \\ -\bar{h}_u \operatorname{sat}(L(t, z(t, x))), & t = t_f, \quad x \in E^6, \end{cases} \tag{5.48} $$

where $t_f = 6.845579$; $L(t, z)$ is given either by (5.41), or by (5.42). The capture zone $C_x$ of these controls is given by the equation (5.24), where $t_0 = 0$, $t_f = 6.845579$, the set $C_z$ is presented in Fig. 5.4 and the function $z(t, x)$ is given by the equation (5.46).
### 5.4.2 Example 2: One Change of Sign of the Determining Function

#### Data of the engagement and the behaviour of the determining function

Consider the engagements (5.7),(2.52) and (2.51),(2.52) with the following set of their data:

$$ 
\begin{array}{l}
V_P(t) = 700 \exp \left(1 + \frac{t}{12}\right) \text{ m/sec, } \quad V_E(t) = 800 \exp \left(1 + \frac{t}{20}\right) \text{ m/sec,} \\
t_0 = 0, \quad \phi_{E0} = \frac{7\pi}{8}, \quad R_0 = 24000 \text{ m, } \quad \tau_P = 0.1 \text{ sec, } \quad \tau_E = 0.12 \text{ sec,} \\
a_{P,c}^{\max}(t) = (20t + 140) \text{ m/sec}^2, \quad a_{E,c}^{\max}(t) = \left(125 \exp \left(\frac{t}{14}\right) + 35\right) \text{ m/sec}^2. \\
\hfill (5.49)
\end{array}
$$

For these data, the equation (2.7) has the unique solution $t_C = t_{C,min} = 5.332963$ sec which, along with (2.8),(2.9),(2.34), yields

$$ \cos(\phi_{PC}) = 0.917359, \quad \sin(\phi_{PC}) = 0.398060, \quad t_f = 5.332963 \text{ sec.} \quad (5.50) $$

Using the equations (5.11),(5.12) and the data (5.49),(5.50), we obtain the determining function in this example

$$ 
\begin{array}{l}
R(t) = 7.698996 \left(125 \exp \left(\frac{t}{14}\right) + 35\right) g_E(t) - 9.173590(20t + 140)g_P(t), \\
g_E(t) = 0.014314 \exp \left(\frac{t - 5.332963}{0.12}\right) \\
\qquad +2.385686 \exp \left(\frac{5.332963 - t}{20}\right) - 2.4, \\
g_P(t) = 0.009917 \exp \left(\frac{t - 5.332963}{0.1}\right) \\
\qquad +1.190083 \exp \left(\frac{5.332963 - t}{12}\right) - 1.2, \\
\hfill (5.51)
\end{array}
$$

where $t \in [0, 5.332963]$.

Graph of this determining function is presented in Fig. 5.9. It is seen that $R(t)$ has a single zero ($t_{s1} = 0.704770$) in the interval $(0, 5.332963)$, and $R(t)$ changes its sign once in this interval. In the interval $(t_{s1}, t_f) = (0.704770, 5.332963)$, the function $R(t)$ is negative.
<figure><img src="image_35.png" /><figcaption>Figure 5.9: Graph of the determining function $R(t)$ in Example 2 (Subsection 5.4.2)</figcaption></figure>

## Decomposition of the pursuit-evasion game's space

Like in Example 5.4.1, we are going to decompose the space of the SPEGBC using the results of Subsection 3.5.3. However, in contrast with Example 5.4.1, this decomposition should be done in accordance with Subcase 1.2, because of the aforementioned behaviour of $R(t)$ in the present example. For a more precise analysis, we should check whether the equation (5.39), where $R(t)$ is given by (5.51), has the unique solution $t = t_{in}$ in the interval $[0, t_{s1})$. Graph of the function $\int_{t_f}^t R(\sigma)d\sigma, t \in [0, t_f]$, depicted in Fig. 5.10, directly shows the nonexistence of such a solution. Thus, to decompose the space of the SPEGBC, we should use the results of Subcase 1.2.2 (see the equations (3.85),((3.77),(3.87) and Fig. 3.6). This yields Fig. 5.11.

## Construction of the robust capture controls and capture zones

Figs. 5.11 and 5.9 immediately show the fulfilment of the conditions of Theorem 5.3.1 and Corollary 5.3.2. Moreover, calculating the expression in the left-hand side of the inequality (5.15) for the data (5.49),(5.50), we obtain the negative number $-584.717030$, meaning that in the present example this inequality is satisfied. Thus, by virtue of any of the assertions, either Theorem 5.3.1 or Corollary 5.3.2 or Lemma 5.3.3, a robust capture control for the engagement (5.7),(2.52) exists in this example. Now, let us construct the capture zones of the controls $u_{sg}^0(t, z)$ and $u_{sg,c}(t, z)$, mentioned in Theorems 5.3.4 and 5.3.5 for the engagement (5.7),(2.52). To do this construction, we should obtain the value $t_{min}$ and the function $z_c(t)$ defined by the equation (5.16).
<figure><img src="image_36.png" /><figcaption>Figure 5.10: Graph of $\int_{t_f}^t R(\sigma)d\sigma$ in Example 2 (Subsection 5.4.2)</figcaption></figure>

From Fig. 5.10, it is seen that the function $\int_{t_f}^t R(\sigma)d\sigma$ is nonnegative in the entire interval $[0, t_f] = [0, 5.332963]$. Thus, $t_{\min} = 0$ and $z_c(t) = \int_{t_f}^t R(\sigma)d\sigma$, $t \in [0, t_f]$. Using the function $z_c(t)$, we construct the capture zone $C_z$ of the control $u_{\text{sg}}^0(t, z)$ mentioned in Theorem 5.3.4. This capture zone is presented in Fig 5.12.

In this example, like in Example 1 (see Subsection 5.4.1), the aforementioned control $u_{\text{sg}}^0(t, z)$ has the form (5.40), but with $t_f = 5.332963$. The control $u_{\text{sg},c}(t, z)$ has the form (5.43), where $t_f = 5.332963$ and

$$L(t, z) = \begin{cases} \frac{z}{z_c(t)}, & t \in [0, t_f), \quad z \in E^1, \\ z, & t = t_f, \quad z \in E^1. \end{cases} \qquad (5.52)$$

Now, proceed to the construction of the function $L_{\text{aux}}(t, z, \lambda)$ and the function $L(t, z)$, which is based on $L_{\text{aux}}(t, z, \lambda)$. The first of these functions is defined by the equations (5.30), where $h_P(t, \phi_{PC})$ and $h_E(t, \phi_{E0})$ are given by the equation (5.9) and the data (5.49)-(5.50); $\lambda \in (0, \lambda_{\inf}(\phi_{PC}, \phi_{E0})) = (0, 1.059429)$ is any prechosen number. The second function is defined by the equations (5.35)-(5.36), i.e., it has the form

$$L(t, z) = L(t, z, \lambda) \triangleq \begin{cases} \gamma_c(\lambda)L_{\text{aux}}(t, z, \lambda), & t \in [0, t_f), \quad z \in E^1, \\ z, & t = t_f, \quad z \in E^1. \end{cases} \qquad (5.53)$$

In Fig. 5.13, the graphs of the function $L_{\text{aux}}(t, z_c(t), \lambda)$ and the function $\gamma_c L_{\text{aux}}(t, z_c(t), \lambda)$ are depicted for different values of $\lambda \in (0, 1.059429)$.

In Fig. 5.14, the trajectories of the scalar pursuit-evasion engagement, governed by the differential equation in (5.7), are presented. All these trajectories start from the point $(t = 0, z = 70) \in C_z$. One of these trajectories (the
<figure><img src="image_37.png" /><figcaption>Figure 5.11: Decomposition of the space of the SPEGBC in Example 2 (Subsection 5.4.2)</figcaption></figure>

solid line) is generated by the robust capture control $u(t) = u_{\text{sg}}^{0}(t, z)$ (see the equation (5.40) with $\alpha_{u}(t) \equiv 1$) and by the evader's control $v(t) = v_{\text{sg}}^{0}(t, z)$ (see the equation (3.74) with $h_{v}(t) = h_{v}(t, \phi_{E0})$ and $\alpha_{v}(t) \equiv 1$). The trajectory, generated by the robust capture control $u(t) = u_{\text{sg,c}}(t, z)$ (with the function $L(t, z)$ given in the equation (5.52)), is depicted by the dashed line. Three other trajectories are generated by the robust capture control $u(t) = u_{\text{sg,c}}(t, z)$ (with the function $L(t, z)$ given in the equation (5.53) and $\lambda = 0.7, 1, 1.04$). It is seen that, starting in the capture zone $C_z$, all these trajectories remain in this set till the end of the engagement. Thus, these trajectories reach the point $(t, z) = (t_f, 0) \in C_z$. It also is seen that the use of the robust capture control $u(t) = u_{\text{sg}}^{0}(t, z)$ yields the sliding mode of the trajectory along the axis $t$ for $t \in [3.415229, t_f]$. The trajectory, generated by the robust capture control $u(t) = u_{\text{sg,c}}(t, z)$ ($L(t, z)$ is given in (5.52)), coincides with the positive boundary of $C_z$ for $t \in [0.859674, t_f]$. The trajectories, generated by the robust capture control $u(t) = u_{\text{sg,c}}(t, z)$ ($L(t, z)$ is given in (5.53), $\lambda = 1$ and $\lambda = 1.04$), coincide with the positive boundary of $C_z$ for $t \in [2.616351, t_f]$ and $t \in [3.073919, t_f]$, respectively. The trajectory, generated by the robust capture control $u(t) = u_{\text{sg,c}}(t, z)$ ($L(t, z)$ is given in (5.53), $\lambda = 0.7$), remains strictly inside $C_z$ for $t \in [0, t_f)$.

In Fig. 5.15, the time realizations of the aforementioned robust capture controls along the corresponding trajectories of the scalar pursuit-evasion engagement are presented. It is seen that the control $u(t) = u_{\text{sg}}^{0}(t, z)$ yields the chattering of its time realization for $t \in [3.415229, t_f]$, while the time realizations of the all considered versions of the control $u(t) = u_{\text{sg,c}}(t, z)$ are
<figure><img src="image_38.png" /><figcaption>Figure 5.12: Capture zone $C_z$ in Example 2 (Subsection 5.4.2)</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>Figure 5.13: Graphs of $L_{\text{aux}}(t, z_c(t), \lambda)$ and $\gamma_c(\lambda)L_{\text{aux}}(t, z_c(t), \lambda)$ in Example 2 (Subsection 5.4.2)</figcaption></figure>

completely chattering free.

**Remark 36.** Similarly to Remark 35, we note the following. Due to the equation (3.74) with $h_v(t) = h_v(t, \phi_{E0})$ and $\alpha_v(t) \equiv 1$, the equation (5.9), and the data (5.49) of this example, we directly obtain that the time realization of the aforementioned evader's control $v_{\text{sg}}^0(t, z)$ along each of the trajectories $z = z(t)$ presented in Fig. 5.14 is $v_{\text{sg}}^0(t, z(t)) \equiv -1$, $t \in [0, t_f]$. Therefore, all the results of Figs. 5.14 and 5.15, remain the same in the case where $v(t) = v_{\text{sg}}^0(t, z)$ is replaced with $v(t) \equiv -1$, while the latter is an admissible evader's control in accordance with Definition 2.6.1.

Now, we are going to construct robust capture controls and corresponding capture zones for the engagement (2.51), (2.52). To carry out this construction, first, we need to know the function $z(t,x)$ (see the equation (5.6)), which derivation is based on the vector-row $\varphi_1(t)$ and the vector-column $f(t, \phi_{\text{PC}}, \phi_{\text{E0}})$. Using the equations (5.4), (5.5) and the data (5.49), (5.50), we
<figure><img src="image_40.png" /><figcaption>Figure 5.14: Trajectories of scalar pursuit-evasion engagement in Example 2 (Subsection 5.4.2)</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>Figure 5.15: Time realizations of the robust capture controls in Example 2 (Subsection 5.4.2)</figcaption></figure>

obtain the entries of the vector-row $\varphi_1(t)$ in the form

$$
\begin{aligned}
\varphi_{11}(t) &= 1, & \varphi_{12}(t) &= 5.332963 - t, & t \in [0, 5.332963], \\
\varphi_{13}(t) &= -11.008308 \left[ \frac{1}{1210} \exp\left(\frac{t-5.332963}{0.1}\right) + \frac{12}{121} \exp\left(\frac{5.332963-t}{12}\right) - 0.1 \right], & t \in [0, 5.332963], \\
\varphi_{14}(t) &= -18.477590 \left[ \frac{1.44}{2012} \exp\left(\frac{t-5.332963}{0.12}\right) + \frac{240}{2012} \exp\left(\frac{5.332963-t}{20}\right) - 0.12 \right], & t \in [0, 5.332963], \\
\varphi_{15}(t) &= 642.1513 \left[ (17.332963 - t) \exp\left(1 + \frac{t}{12}\right) - 50.872385 \right], & t \in [0, 5.332963],
\end{aligned}
$$
$$ \varphi_{16}(t) = 739.1036 \left[ \begin{aligned} & (25.332963 - t) \exp \left( 1 + \frac{t}{20} \right) \\ & -70.978742 \end{aligned} \right], \quad t \in [0, 5.332963]. \tag{5.54} $$

Using the data (5.49), (5.50), we obtain the second entry of the vector-column $f(t, \phi_{PC}, \phi_{E0})$ (see the equations (2.47), (2.48)) in the form

$$ f_2(t, \phi_{PC}, \phi_{E0}) = -23.220167 \exp \left( 1 + \frac{t}{12} \right) + 15.307336 \exp \left( 1 + \frac{t}{20} \right), \quad t \in [0, 5.332963]. \tag{5.55} $$

Thus, the aforementioned function $z(t, x)$ has the form

$$ \begin{aligned} z(t, x) = & x_1 + (5.332963 - t)x_2 + \varphi_{13}(t)x_3 + \varphi_{14}(t)x_4 + \varphi_{15}(t)x_5 \\ & + \varphi_{16}(t)x_6 - 278.642\left[(t - 17.332963)\exp\left(1 + \frac{t}{12}\right) + 50.872385\right] \\ & + 306.14672\left[(t - 25.332963)\exp\left(1 + \frac{t}{20}\right) + 70.978742\right], \quad t \in [0, 5.332963]. \end{aligned} \tag{5.56} $$

**Remark 37.** Let us consider the controls $u^0(t, x)$ and $u_c(t, x)$ of the form (5.47) and (5.48), respectively, while with $t_f = 5.332963$, the function $L(t, z)$ given either by (5.52) or by (5.53), and the function $z(t, x)$ given by (5.56). Similarly to Example 1 (see Subsection 5.4.1), these controls are robust capture controls for the engagement (2.51), (2.52). The capture zone $C_x$ of these controls is given by the equation (5.24), where $t_0 = 0$, $t_f = 5.332963$, the set $C_z$ is presented in Fig. 5.12 and the function $z(t, x)$ is given by the equation (5.56).

### 5.4.3 Evaluation of the Robust Capture Controls in Numerical Simulations

In this subsection, the robust capture controls obtained in Subsections 5.4.1 and 5.4.2 are evaluated by numerical simulations subject to the following realistic assumptions:

* nonlinear engagement model;

* random wind disturbance.
Due to these assumptions and the results of Chapter 2 (see Sections 2.2-2.5), the pursuit-evasion engagement of two flying vehicles is described by the following set of eight differential equations subject to starting engagement's position:

$$
\begin{align}
\frac{dx_P(t)}{dt} &= V_P(t) \cos \phi_P(t) + W_{P,x}(t), & x_P(t_{\text{st}}) &= x_{P,\text{st}}, \nonumber \\
\frac{dy_P(t)}{dt} &= V_P(t) \sin \phi_P(t) + W_{P,y}(t), & y_P(t_{\text{st}}) &= y_{P,\text{st}}, \nonumber \\
\frac{d\phi_P(t)}{dt} &= \frac{a_P(t)}{V_P(t)}, & \phi_P(t_{\text{st}}) &= \phi_{P,\text{st}}, \nonumber \\
\frac{da_P(t)}{dt} &= \frac{a_{P,c}^{\max}(t)u(t) - a_P(t)}{\tau_P}, & a_P(t_{\text{st}}) &= a_{P,\text{st}}, \nonumber \\
\frac{dx_E(t)}{dt} &= V_E(t) \cos \phi_E(t) + W_{E,x}(t), & x_E(t_{\text{st}}) &= x_{E,\text{st}}, \nonumber \\
\frac{dy_E(t)}{dt} &= V_E(t) \sin \phi_E(t) + W_{E,y}(t), & y_E(t_{\text{st}}) &= y_{E,\text{st}}, \nonumber \\
\frac{d\phi_E(t)}{dt} &= \frac{a_E(t)}{V_E(t)}, & \phi_E(t_{\text{st}}) &= \phi_{E,\text{st}}, \nonumber \\
\frac{da_E(t)}{dt} &= \frac{a_{E,c}^{\max}(t)v(t) - a_E(t)}{\tau_E}, & a_E(t_{\text{st}}) &= a_{E,\text{st}}, \tag{5.57}
\end{align}
$$

where

$$
W_P(t) = \operatorname{col}(W_{P,x}(t), W_{P,y}(t)) \qquad (5.58)
$$

and

$$
W_E(t) = \operatorname{col}(W_{E,x}(t), W_{E,y}(t)) \qquad (5.59)
$$

are time-varying wind's velocity vectors, influencing on the pursuer and the
evader, respectively.

**Remark 38.** In this subsection, the simulations are carried out subject to the condition that the starting engagement's position belongs to the capture zone constructed in Subsections 5.4.1 and 5.4.2.

Remember that in the equation (5.57) and in what follows of this subsec-
tion $P(t) = (x_P(t), y_P(t))$ and $E(t) = (x_E(t), y_E(t))$ are the current positions
of the pursuer and the evader, respectively, in the plane $XOY$ (the ori-
gin $O$ of this coordinate system coincides with the initial pursuer's position
$P_0 = P(0)$); $\phi_P(t)$, $(\phi_E(t))$ is the current pursuer's (evader's) aspect angle,
i.e., the angle between the current pursuer's (evader's) velocity vector $\mathbf{V}_P(t)$
$(V_E(t))$ and the X-axis; $V_P(t)$ and $V_E(t)$ are the magnitudes of the current velocity vectors of the pursuer and the evader, respectively; $a_P(t)$ and $a_E(t)$ are the magnitudes of the current lateral acceleration vectors $a_P(t)$ and $a_E(t)$ of the pursuer and the evader, respectively; $a_{P,c}^{\max}(t)$ ($a_{E,c}^{\max}(t)$) is the current maximum of the absolute value of the pursuer's (the evader's) lateral acceleration command; $u(\cdot)$ and $v(\cdot)$ are the controls of the pursuer and the evader, respectively. More details can be found in the aforementioned sections of Chapter 2.

In the model of the wind's velocity vector, influencing on each vehicle, it is assumed that its magnitude and the direction's angle, are combined from constant and time-varying parts. The constant parts are the wind's magnitude $A_C$ and the direction's angle $\psi_C$ at the vehicles' collision point. The time-varying parts are considerable when the vehicles are far from the collision point, while these parts decrease when the vehicles approach this point becoming zero at this point. Let $A_{P0}$ and $\psi_{P0}$ be the magnitude and the angle of the wind, influencing on the pursuer at the beginning of the engagement ($t = t_{st}$); $A_{E0}$ and $\psi_{E0}$ be the magnitude and the angle of the wind, influencing on the evader at $t = t_{st}$. Consider the following values:

$$ \Delta A_k \triangleq A_{k0} - A_C, \quad \Delta \psi_k \triangleq \psi_{k0} - \psi_C, \quad k = P, E. \qquad (5.60) $$

Using the aforementioned assumption on the behaviour of the magnitudes and the direction angles of the wind, as well as the additional assumption that the magnitudes and the direction angles of the wind influencing on the vehicles linearly depend on $\Delta A_k$ and $\Delta \psi_k$, ($k = P, E$), we represent the time-dependent magnitudes and the direction angles of the wind as:

$$ A_k(t) = A_C + \frac{t_{go}(t)}{t + t_{go}(t)} \Delta A_k, \quad k = P, E, \quad t \ge t_{st}, \qquad (5.61) $$

$$ \psi_k(t) = \psi_C + \frac{t_{go}(t)}{t + t_{go}(t)} \Delta \psi_k, \quad k = P, E, \quad t \ge t_{st}, \qquad (5.62) $$

where $t_{go}(t) = t_{go}(P(t), E(t), \phi_P(t), \phi_E(t), t)$ is the current time-to-go, i.e., the estimated duration of the engagement from its current kinematics position $(P(t), E(t), \phi_P(t), \phi_E(t), t)$ to the collision point.

**Remark 39.** For any current kinematics position $(P(t), E(t), \phi_P(t), \phi_E(t), t)$, $t \ge 0$ of the engagement, the time-to-go $t_{go}(t)$ is calculated in the following way. First, we calculate the functions

$$ R_x(t) \triangleq x_E(t) - x_P(t), \quad R_y(t) \triangleq y_E(t) - y_P(t), \quad t \ge t_{st}, \qquad (5.63) $$
$$R(t) \triangleq \sqrt{(R_x(t))^2 + (R_y(t))^2}, \quad t \ge t_{\text{st}}. \qquad (5.64)$$

Then, using this functions and extending the results of Section 2.3 to the case where $y_P(t) \neq 0$, $y_E(t) \neq 0$, we construct the following nonlinear algebraic equation with respect to the unknown $t_{\text{go}}$:

$$\left[ (R(t))^2 + 2(R_x(t) \cos(\phi_E(t)) + R_y(t) \sin(\phi_E(t))) \int_t^{t+t_{\text{go}}} V_E(\sigma) d\sigma + \left( \int_t^{t+t_{\text{go}}} V_E(\sigma) d\sigma \right)^2 \right]^{1/2} . \qquad (5.65)$$

The smallest nonnegative solution $t_{\text{go}}$ of this equation is the sought time-to-go $t_{\text{go}}(t)$. Alternatively, the time-to-go $t_{\text{go}}(t)$ can be calculated as follows:

$$t_{\text{go}}(t) = \left| \frac{R(t)}{dR(t)/dt} \right|. \qquad (5.66)$$

Thus, using the equations (5.58)-(5.59) and (5.61)-(5.62), we obtain the components of the wind's velocity vectors in the x- and y-directions

$$W_{k,x}(t) = A_k(t) \cos \psi_k(t), \quad k = P, E, \quad t \ge t_{\text{st}}, \qquad (5.67)$$

$$W_{k,y}(t) = A_k(t) \sin \psi_k(t), \quad k = P, E, \quad t \ge t_{\text{st}}. \qquad (5.68)$$

In this simulation study, the magnitude and the direction of the wind are modeled following the results of the paper [15]. Namely, the values $A_C$, $\Delta A_P$ and $\Delta A_E$ are chosen as normally distributed random ones:

$$A_C \sim N(\mu_{A_C}, \sigma_{A_C}), \quad \Delta A_k \sim N(\mu_{\Delta A_k}, \sigma_{\Delta A_k}), \quad k = P, E. \qquad (5.69)$$

For the values $\psi_C$, $\Delta\psi_P$ and $\Delta\psi_E$, the von Mises distribution (see e.g. the book [8]) is employed. This distribution is an analogue of the Gaussian distribution on the circle. Its probability density function is

$$f_{\text{VM}}(\psi; \mu, \kappa) = \frac{\exp(\kappa(\mu - \psi))}{2\pi I_0(\kappa)}, \qquad (5.70)$$

where $I_0(\kappa)$ is the modified Bessel function of the first kind of order 0, $\mu$ is the mean, $\kappa$ characterizes the concentration of the random value around $\mu$.
In what follows of this subsection, we denote the random value $\psi$, distributed in the accordance with (5.70), as $\psi \sim VM(\mu, \kappa)$, and assume

$$ \psi_C \sim VM(\mu_{\psi_C}, \kappa_{\psi_C}), \quad \Delta\psi_k \sim VM(\mu_{\Delta\psi_k}, \kappa_{\Delta\psi_k}), \quad k = P, E. \quad (5.71) $$

In the simulation, the pursuer uses the robust capture controls, derived in Subsection 5.4.1 (Example 1, the equations (5.46), (5.47), (5.48)) and Subsection 5.4.2 (Example 2, Remark 37). For any current position $(P(t), E(t), a_P(t), a_E(t), \phi_P(t), \phi_E(t))$, $t \ge t_{st}$ of the engagement, the entries of the vector $x$, appearing in the aforementioned robust capture controls, are defined in Section 2.4 (see Fig. 2.2 and the equation (2.35)), while the value $t_f$, appearing in these controls, is replaced with the sum $t + t_{go}(t)$.

The evader employs the feedback control $v_{sg}^0(t, z)$ of the form (3.74) with $h_v(t) = h_v(t, \phi_{E0})$, $\alpha_v(t) \equiv 1$ and $t_0 = t_{st}$. This control is the worst case evader's control from the pursuer's viewpoint.

**Remark 40.** Due to the nonlinear kinematics models of the pursuer and the evader and the presence of the wind influencing on the vehicles (see the equation (5.57)), the condition $P(t) = E(t)$ cannot, in general, be fulfilled for any $t \ge 0$ even if the engagement starts in the capture zone constructed in Sections 5.3 and 5.4. Therefore, in the simulation, we calculate the local minimum $R_{f,cond}$ of the function $R(t)$ subject to the conditions $\{t \ge 0\} \cup \{R(t) \le \tilde{R}\}$, which corresponds to the smallest minimum point $t_{f,cond}$. In this calculation, the function $R(t)$ is given by (5.64); $\tilde{R} > 0$ is a properly prechosen (sufficiently small from the practical viewpoint) value. Thus, in the simulation, the fulfillment of the equality $R(t) = R_{f,cond}$ means the end of the engagement and it is called the conditional capture. The value $R_{f,cond}$ itself represents the miss distance in the engagement, while the value $t_{f,cond}$ is the time of the conditional capture (the final time of the engagement).

Below, the results of numerical simulation for the robust capture controls derived in Subsections 5.4.1 and 5.4.2 are presented.

## Simulation Results for the Robust Capture Controls of Subsection 5.4.1

The starting position of the engagement in this simulation is the following: $t_{st} = 4$ [sec], $x_{P,st} = 7650.932$ [m], $y_{P,st} = 4880.074$ [m], $\phi_{P,st} = 0.7006$ [rad], $a_{P,st} = 197.625$ [m/s²], $x_{E,st} = 21297.281$ [m], $y_{E,st} = 5701.310$ [m], $\phi_{E,st} = 2.4821$ [rad], $a_{E,st} = -115.871$ [m/s²].

In Tables 5.1 and 5.2, the wind amplitude and angle data, used in the simulation, are presented.
<table><caption>Table 5.1: Wind parameters (amplitude)</caption><thead><tr><th></th><th>A<sub>C</sub> [m/s]</th><th>ΔA<sub>P</sub> [m/s]</th><th>ΔA<sub>E</sub> [m/s]</th></tr></thead><tbody><tr><th>μ</th><td>25</td><td>30</td><td>15</td></tr><tr><th>σ</th><td>1.5</td><td>3</td><td>2</td></tr></tbody></table>

<table><caption>Table 5.2: Wind parameters (angle)</caption><thead><tr><th></th><th>ψ<sub>C</sub> [rad]</th><th>Δψ<sub>P</sub> [rad]</th><th>Δψ<sub>E</sub> [rad]</th></tr></thead><tbody><tr><th>μ</th><td>0</td><td>0.7006 + π/2</td><td>2.4821 − π/2</td></tr><tr><th>κ</th><td>π/10</td><td>π/10</td><td>π/10</td></tr></tbody></table>

**Remark 41.** Based on the data of Table 5.2 and on the equation (5.60), one can conclude the following. The direction of the wind, influencing on the pursuer at the beginning of the engagement, is perpendicular to the pursuer's velocity vector and this wind rotates the pursuer's velocity vector counterclockwise. Furthermore, the direction of the wind, influencing on the evader at the beginning of the engagement, is perpendicular to the direction of the evader's velocity vector and this wind rotates the evader's velocity vector clockwise. Thus, both rotations are opposite to each other. Hence, these winds' influences maximally increase the distance between the pursuer and the evader (the worst case winds from the pursuer's viewpoint), which complicates the capture considerably.

In Fig. 5.16, the cumulative distribution function of the miss distance ("miss distance" → "probability") is shown for all the pursuer's robust capture controls designed in Subsection 5.4.1. In Table 5.3, the miss distance characteristics for these pursuer's robust capture controls are presented. In this table, $R_{av}$ denotes the average miss distance in the simulation with 500 Monte Carlo runs, $R_{pr}$ denotes the "pr" miss distance percentile. It is seen that the robust capture control $u = u_{sg}(t, z)$ (the bang-bang control) yields the best miss distance results. The control $u = u_{sg,c}(t, z)$, given either by (5.41) or by (5.42), yields the larger miss distance than $u = u_{sg,c}(t, z)$. Nevertheless, the largest miss distance in Fig. 5.16 and in Table 5.3 is less than 30 [cm]. In a practical situation such a result allows to the pursuer's vehicle with any of the controls ($u = u_{sg}(t, z)$ or $u = u_{sg,c}(t, z)$) to shoot down the evader's vehicle by a direct hit even with the probability 0.95. However, the bang-bang control $u = u_{sg}(t, z)$ yields the huge chattering as it is shown in Subsection 5.4.1 (see Fig. 5.8). This chattering can damage or even destroy the pursuer's vehicle, which will not allow to the pursuer to carry out its mission. Taking into account these observations, one can conclude that it is
<figure><img src="image_42.png" /><figcaption>Figure 5.16: Miss distance cumulative distribution function for $u_{\text{sg}}^{0}(t, z)$, $u_{\text{sg,c}}^{0}(t, z)$ with $L(t, z)$ (5.41) and $u_{\text{sg,c}}^{0}(t, z)$ with $L(t, z)$ (5.42)</figcaption></figure>

preferable for the pursuer to use of the control $u = u_{\text{sg,c}}^{0}(t, z)$ given either by (5.41) or by (5.42).

<table><caption>Table 5.3: Miss distance characteristics</caption><thead><tr><th></th><th>R<sub>av</sub> [m]</th><th>R<sub>95</sub> [m]</th><th>R<sub>90</sub> [m]</th><th>R<sub>85</sub> [m]</th><th>R<sub>80</sub> [m]</th></tr></thead><tbody><tr><td>u = u<sup>0</sup><sub>sg</sub>(t, z)</td><td>0.0941</td><td>0.1599</td><td>0.1551</td><td>0.1500</td><td>0.1431</td></tr><tr><td>u = u<sup>0</sup><sub>sg,c</sub>(t, z) (5.41)</td><td>0.2279</td><td>0.2428</td><td>0.2404</td><td>0.2391</td><td>0.2372</td></tr><tr><td>u = u<sup>0</sup><sub>sg,c</sub>(t, z) (5.42)</td><td>0.2826</td><td>0.2936</td><td>0.2911</td><td>0.2903</td><td>0.2887</td></tr></tbody></table>

## Simulation Results for the Robust Capture Controls of Subsection 5.4.2

The starting position of the engagement in this simulation is the following: $t_{\text{st}} = 0$ [sec], $x_{P,\text{st}} = 0$ [m], $y_{P,\text{st}} = 0$ [m], $\phi_{P,\text{st}} = 0.4034$ [rad], $a_{P,\text{st}} = 0$ [m/s²], $x_{E,\text{st}} = 30000$ [m], $y_{E,\text{st}} = 0$ [m], $\phi_{E,\text{st}} = 7\pi/8$ [rad], $a_{E,\text{st}} = 0$ [m/s²].

In Tables 5.4 and 5.5, the wind amplitude and angle data, used in the simulation, are presented.

**Remark 42.** As it is aforementioned, the value 0.4034 is the pursuer's angle $\phi_{P,\text{st}}$ in the starting position of the engagement, while the evader's angle $\phi_{E,\text{st}}$ in this position is $7\pi/8$. Thus, based on the data of Table 5.5 and on the equation (5.60), one can conclude the following. The direction of the wind, influencing on the pursuer at the beginning of the engagement, is perpendicular to the pursuer's velocity vector and this wind rotates the pursuer's velocity
<table><caption>Table 5.4: Wind parameters (amplitude)</caption><thead><tr><th></th><th>A<sub>C</sub> [m/s]</th><th>ΔA<sub>P</sub> [m/s]</th><th>ΔA<sub>E</sub> [m/s]</th></tr></thead><tbody><tr><th>μ</th><td>25</td><td>30</td><td>15</td></tr><tr><th>σ</th><td>1.5</td><td>3</td><td>2</td></tr></tbody></table>

<table><caption>Table 5.5: Wind parameters (angle)</caption><thead><tr><th></th><th>ψ<sub>C</sub> [rad]</th><th>Δψ<sub>P</sub> [rad]</th><th>Δψ<sub>E</sub> [rad]</th></tr></thead><tbody><tr><th>μ</th><td>0</td><td>0.4034 + π/2</td><td>3π/8</td></tr><tr><th>κ</th><td>π/10</td><td>π/10</td><td>π/10</td></tr></tbody></table>

vector counterclockwise. Furthermore, the direction of the wind, influencing on the evader at the beginning of the engagement, is perpendicular to the direction of the evader's velocity vector and this wind rotates the evader's velocity vector clockwise. Thus, both rotations are opposite to each other. Hence, these winds' influences maximally increase the distance between the pursuer and the evader (the worst case winds from the pursuer's viewpoint), which complicates the capture considerably.

<figure><img src="image_43.png" /><figcaption>Figure 5.17: Miss distance cumulative distribution function for $u_{sg,c}^0(t, z)$ with $L(t, z)$ (5.53)</figcaption></figure>

In Fig. 5.17, the cumulative distribution function of the miss distance ("miss distance" → "probability") is shown for the pursuer's robust capture control $u_{sg,c}^0(t, z)$ with the function $L(t, z)$ given by the equation (5.53) for various values of $\lambda$. The number of Monte Carlo runs in this simulation is 500.
<figure><img src="image_44.png" /><figcaption>Figure 5.18: Miss distance cumulative distribution function for $u_{\text{sg}}^{0}(t, z)$, $u_{\text{sg,c}}^{0}(t, z)$ with $L(t, z)$ (5.52) and $u_{\text{sg,c}}^{0}(t, z)$ with $L(t, z)$ (5.53), $\lambda = 1$</figcaption></figure>

It is seen that, with the probability from 0.5 to 1, the control $u_{\text{sg,c}}^{0}(t, z)$ for $\lambda = 1$ yields the smaller miss distance than such controls for $\lambda = 0.7$ and $\lambda = 1.04$. In Fig. 5.18, the cumulative distribution function of the miss distance is shown for the pursuer's robust capture controls designed in Subsection 5.4.2. It is seen that, with the probability from 0.45 to 1 (excepting the interval (0.85, 0.9]), the control $u_{\text{sg,c}}^{0}(t, z)$ with $L(t, z)$ from (5.53) and $\lambda = 1$ yields the smaller miss distance than the other two controls. In the probability interval (0.85, 0.9], the miss distance, yielding by the control $u_{\text{sg,c}}^{0}(t, z)$ with $L(t, z)$ from (5.52), is equal or a bit smaller than the miss distance yielding by the control $u_{\text{sg,c}}^{0}(t, z)$ with $L(t, z)$ from (5.53) and $\lambda = 1$.

In Table 5.6, the miss distance characteristics for all the pursuer's robust capture controls, designed in Subsection 5.4.2, are presented. In this table, $R_{\text{av}}$ denotes the average miss distance in the simulation with 500 Monte Carlo runs, $R_{\text{pr}}$ denotes the "pr" miss distance percentile. It is seen that the robust capture control $u = u_{\text{sg,c}}^{0}(t, z)$ with the function $L(t, z)$ given by the equation (5.53) and $\lambda = 1.00$ yields the best results for $R_{\text{av}}$ and almost all $R_{\text{pr}}$ (excepting $R_{90}$). Moreover, the control $u = u_{\text{sg,c}}^{0}(t, z)$ with the function $L(t, z)$ given by either the equation (5.52) or the equation (5.53) provides better results than the bang-bang control $u = u_{\text{sg}}^{0}(t, z)$ which yields the huge chattering.
<table><caption>Table 5.6: Miss distance characteristics</caption><thead><tr><th></th><th>R<sub>av</sub> [m]</th><th>R<sub>95</sub> [m]</th><th>R<sub>90</sub> [m]</th><th>R<sub>85</sub> [m]</th><th>R<sub>80</sub> [m]</th></tr></thead><tbody><tr><td>u = u<sup>0</sup><sub>sg</sub>(t, z)</td><td>0.1063</td><td>0.1814</td><td>0.1765</td><td>0.1718</td><td>0.1665</td></tr><tr><td>u = u<sup>0</sup><sub>sg,c</sub>(t, z) (5.52)</td><td>0.0880</td><td>0.1493</td><td>0.1429</td><td>0.1381</td><td>0.1319</td></tr><tr><td>u = u<sup>0</sup><sub>sg,c</sub>(t, z) (5.53), λ = 0.70</td><td>0.0900</td><td>0.1502</td><td>0.1462</td><td>0.1430</td><td>0.1384</td></tr><tr><td>u = u<sup>0</sup><sub>sg,c</sub>(t, z) (5.53), λ = 1.00</td><td>0.0862</td><td>0.1487</td><td>0.1441</td><td>0.1374</td><td>0.1302</td></tr><tr><td>u = u<sup>0</sup><sub>sg,c</sub>(t, z) (5.53), λ = 1.04</td><td>0.0898</td><td>0.1496</td><td>0.1455</td><td>0.1403</td><td>0.1354</td></tr></tbody></table>

# 5.5 Proofs of Theorems 5.3.1-5.3.5, Lemmas 5.3.3-5.3.10 and Proposition 5.3.12

## 5.5.1 Proof of Theorem 5.3.1

*Sufficiency.* Let the inclusion (5.13) is valid. Then, due to the construction of the regular and singular regions of the SPEGBC (see Subsection 3.5.3), the point $(t_f, 0)$ belongs to the positive boundary $z = z^+(t)$ of the regular region, i.e., $z^+(t_f) = 0$. Furthermore, there exists a simply connected set $\tilde{\mathcal{R}}_s \subseteq \mathcal{R}_s$ such that $(t_f, 0) \in \text{clo}(\tilde{\mathcal{R}}_s)$. Moreover, there exists a sequence of points $\{(t_i, z_i)\}_{i=1}^{+\infty}$ such that $(t_i, z_i) \in \tilde{\mathcal{R}}_s, i = 1, 2, \dots$, and $\lim_{i \to +\infty} (t_i, z_i) = (t_f, 0)$. Due to these observations and Theorem 3.5.13, we obtain the following limit equality for the value of the SPEGBC:

$$ \lim_{i \to +\infty} J_z^0(t_i, z_i) = \lim_{i \to +\infty} \left( z^+(t_i) + \int_{t_i}^{t_f} R(\sigma) d\sigma \right) = 0. \quad (5.72) $$

Since the set $\tilde{\mathcal{R}}_s$ is a simply connected part of the singular region $\mathcal{R}_s$, Corollary 3.5.14 yields the equality $J_z^0(t_i, z_i) = J_z^0(t_j, z_j)$ for all $i \in \{1, 2, \dots\}$, $j \in \{1, 2, \dots\}$. This equality, along with the limit equality (5.72) directly implies that

$$ J_z^0(\bar{t}, \bar{z}) = 0 \quad \forall (\bar{t}, \bar{z}) \in \tilde{\mathcal{R}}_s. \quad (5.73) $$

Since $J_z^0(\bar{t}, \bar{z})$ is the SPEGBC value, then due to the equations (5.8), (5.73), Theorem 3.5.13 and Remark 30, we directly can conclude the existence of a robust capture control for the engagement (5.7), (2.52). This completes the proof of the sufficiency.

*Necessity.* Let us assume the existence of a robust capture control $u_{\text{rcc}}(t, z)$, $(t, z) \in [t_0, t_f] \times E^1$ for the engagement (5.7), (2.52). Due to Remark 30, this means the existence of a point $(\bar{t}, \bar{z}) \in [t_0, t_f) \times E^1$ such that, for any admissible (in the sense of Definition 2.6.1) evader's control $v(t)$, the solution
$z(t) = z(t; \bar{t}, \bar{z}, v(\cdot))$ of the initial-value problem

$$ \frac{dz}{dt} = h_u(t, \phi_{PC})u_{rcc}(t, z) + h_v(t, \phi_{E0})v(t), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z} \quad (5.74) $$

exists, is unique and satisfies the limit equality

$$ z(t_f) \stackrel{\Delta}{=} \lim_{t \to t_f - 0} z(t) = 0, \quad (5.75) $$

while the time realization $u_{rcc}(t) \stackrel{\Delta}{=} u_{rcc}(t, z(t))$ of the control $u_{rcc}(t, z)$ along this solution satisfies the inequality

$$ |u_{rcc}(t)| \le 1, \quad t \in [\bar{t}, t_f]. \quad (5.76) $$

Since $z(t)$ is the unique solution of the initial-value problem (5.74) and $u_{rcc}(t)$ is the time realization of the control $u_{rcc}(t, z)$ along this solution, then $z(t)$ also is the unique solution of the initial-value problem

$$ \frac{dz}{dt} = h_u(t, \phi_{PC})u_{rcc}(t) + h_v(t, \phi_{E0})v(t), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}. \quad (5.77) $$

Let us consider the initial-value problem

$$ \frac{dz}{dt} = h_u(t, \phi_{PC})u_{sg}^0(t, z) + h_v(t, \phi_{E0})v(t), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}, \quad (5.78) $$

where $u_{sg}^0(t, z)$ is of the form (3.73) with $h_u(t) = h_u(t, \phi_{P0})$ and $0 < |\alpha_u(t)| \le 1$ meaning that

$$ |u_{sg}^0(t, z)| \le 1, \quad (t, z) \in [\bar{t}, t_f] \times E^1. \quad (5.79) $$

Moreover, due to the inequality (5.76) and the aforementioned form of $u_{sg}^0(t, z)$, we have the inequalities

$$ \begin{aligned} h_u(t, \phi_{PC})u_{rcc}(t) &\ge h_u(t, \phi_{PC})u_{sg}^0(t, z), \quad t \in [\bar{t}, t_f], \quad z > 0, \\ h_u(t, \phi_{PC})u_{rcc}(t) &\le h_u(t, \phi_{PC})u_{sg}^0(t, z), \quad t \in [\bar{t}, t_f], \quad z < 0. \end{aligned} \quad (5.80) $$

The initial-value problem (5.78) has the unique solution (the continuous one) $z = z^0(t)$ in the sense of the particular case of the Krasovskii's constructive motion [9] (see Remark 6).

Comparing the initial-value problems (5.77) and (5.78), and taking into account the inequalities (5.80), we directly can conclude that

$$ |z^0(t)| \le |z(t)|, \quad t \in [\bar{t}, t_f], $$
yielding, along with (5.75), that

$$z^0(t_f) = 0. \qquad (5.81)$$

Note that this equality is valid for any evader's control $v(t)$ satisfying the inequality $|v(t)| \le 1$.

Now, we consider the following initial-value problem:

$$\frac{dz}{dt} = h_u(t, \phi_{PC})u_{sg}^0(t, z) + h_v(t, \phi_{E0})v_{sg}^0(t, z), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}, \qquad (5.82)$$

where $u_{sg}^0(t)$ is the same as in (5.78); $v_{sg}^0(t, z)$ is of the form (3.74) with $h_v(t) = h_v(t, \phi_{E0})$ and $0 < |\alpha_v(t)| \le 1$ meaning that

$$|v_{sg}^0(t, z)| \le 1, \quad (t, z) \in [\bar{t}, t_f] \times E^1. \qquad (5.83)$$

The initial-value problem (5.82) has the unique solution (the continuous one) $z = z_{sg}^0(t)$ in the sense of the particular case of the Krasovskii's constructive motion [9] (see Remark 20). By virtue of Theorem 3.5.13, this solution is the saddle-point trajectory in feedback controls of the SPEGBC starting at $(\bar{t}, \bar{z})$ (see Section 3.5, Definition 3.5.8). Therefore, $|z_{sg}^0(t_f)|$ is the value of the SPEGBC with the starting point $(\bar{t}, \bar{z})$.

Let $v_{sg}^0(t) \triangleq v_{sg}^0(t, z_{sg}^0(t))$ be the time realization of the evader's control $v_{sg}^0(t, z)$ along the solution $z_{sg}^0(t)$. Due to the inequality (5.83), we have

$$|v_{sg}^0(t)| \le 1, \quad t \in [\bar{t}, t_f]. \qquad (5.84)$$

Moreover, since $z = z_{sg}^0(t)$ is the unique solution of the initial-value problem (5.82), it is the unique solution of the following initial-value problem in the sense of the particular case of the Krasovskii's constructive motion [9] (see Remark 6):

$$\frac{dz}{dt} = h_u(t, \phi_{PC})u_{sg}^0(t, z) + h_v(t, \phi_{E0})v_{sg}^0(t), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}. \qquad (5.85)$$

Hence, taking into account the inequality (5.84) and the property (5.81) of the solution to the problem (5.78), we have immediately that $z_{sg}^0(t_f) = 0$. Thus, the saddle-point trajectory in feedback controls of the SPEGBC, starting at $(\bar{t}, \bar{z})$, ends at $(t_f, 0)$. Therefore, the value of the SPEGBC with the starting point $(\bar{t}, \bar{z})$ equals zero. This observation, along with the decomposition of the SPEGBC space (see Subsection 3.5.3), directly yields that the point $(t, z) = (t_f, 0)$ belongs simultaneously to the positive and negative boundaries $z^+(t)$ and $z^-(t)$ of the SPEGBC regular region, i.e., $z^+(t_f) = z^-(t_f) = 0$. Therefore, the point $(t, z) = (t_f, 0)$ necessarily belongs to the closure of the singular region $\mathcal{R}_s$ of the SPEGBC, i.e., the equation (5.13) is valid. This completes the proof of the necessity. Thus, the theorem is proven.
### 5.5.2 Proof of Lemma 5.3.3

First of all, let us note the following. Although $R(t)$ is a continuous function in the interval $[t_0, t_f]$, we cannot check the validity of (5.14) by the sign of $R(t_f)$, because $R(t_f) = 0$. Therefore, we should check the validity of (5.14) by a proper sign of $d^k R(t_f)/dt^k \neq 0$, where $k > 0$ is the smallest integer. Let calculate $dR(t_f)/dt$. For this purpose, due to (5.11), we should calculate $dg_l(t)/dt$, ($l = P, E$). Using (5.12), we directly have

$$ \frac{dg_l(t)}{dt} = \frac{1}{V_l(t)} \int_{t_f}^{t} V_l(\sigma) d\sigma + \frac{1}{\tau_l} \int_{t_f}^{t} \left[ \exp\left(\frac{t-\xi}{\tau_l}\right) \frac{1}{V_l(\xi)} \int_{t_f}^{\xi} V_l(\sigma) d\sigma \right] d\xi, \quad t \in [t_0, t_f], \quad l = P, E. \quad (5.86) $$

Therefore, $dg_l(t_f)/dt = 0$, ($l = P, E$). Moreover, (5.12) yields that $g_l(t_f) = 0$, ($l = P, E$). Since $g_l(t)$ and $dg_l(t)/dt$, ($l = P, E$) become zero at $t = t_f$, then due to (5.11), $dR(t_f)/dt = 0$. The latter means that $k > 1$. Proceed with the calculation of $d^2R(t_f)/dt^2$. This calculation needs to know $d^2g_l(t)/dt^2$, ($l = P, E$). Using (5.86), we obtain the following expression in the left-hand vicinity of $t = t_f$ mentioned in the lemma:

$$ \frac{d^2 g_l(t)}{dt^2} = -\frac{dV_l(t)}{dt} \int_{t_f}^{t} V_l(\sigma) d\sigma + 1 + \frac{1}{\tau_l V_l(t)} \int_{t_f}^{t} V_l(\sigma) d\sigma + \frac{1}{\tau_l^2} \int_{t_f}^{t} \left[ \exp\left(\frac{t-\xi}{\tau_l}\right) \frac{1}{V_l(\xi)} \int_{t_f}^{\xi} V_l(\sigma) d\sigma \right] d\xi, \quad t \in [t_0, t_f], \quad l = P, E, $$

which yields $d^2g_l(t_f)/dt^2 = 1$, ($l = P, E$). Using the latter, as well as the equalities $g_l(t_f) = 0$ and $dg_l(t_f)/dt = 0$, ($l = P, E$), we obtain by a direct calculation that

$$ \frac{d^2 R(t_f)}{dt^2} = \frac{|\cos(\phi_{E0})|}{\tau_E} a_{E,c}^{\max}(t_f) - \frac{|\cos(\phi_{PC})|}{\tau_P} a_{P,c}^{\max}(t_f). $$

This implies, due to (5.15), $d^2R(t_f)/dt^2 < 0$. Since $R(t)$ is a twice differentiable function in some left-hand vicinity of $t = t_f$ including $t = t_f$ and $R(t_f) = 0$, $dR(t_f)/dt = 0$, there exists a left-hand vicinity of $t = t_f$ not including $t = t_f$, in which $R(t)$ is negative. The latter means the validity of the inequality (5.14) and, therefore, the existence of a robust capture control for the engagement (5.7),(2.52). Thus, the lemma is proven.

### 5.5.3 Proof of Theorem 5.3.4

First of all, let us observe the following. Due to the results of Subsection 3.5.3 (Decomposition of the game space), the curves $z = z_c(t)$ and $z = -z_c(t)$,
$t \in [t_{\min}, t_f]$ are segments of the positive and negative boundaries of the SPEGBC regular region. Therefore, the interior $\text{int}(C_z)$ of the set $C_z$ is a part of the SPEGBC singular region. Moreover, $\text{int}(C_z)$ is the maximal part of the SPEGBC singular region which closure is a simply connected set and this set contains the point $(t_f, 0)$. Using this observation, the equation (5.16) and Theorem 3.5.13 (see Subsection 3.5.3), we obtain the value $J_z^0(\bar{t}, \bar{z})$ of the SPEGBC for all its starting points $(\bar{t}, \bar{z}) \in C_z$

$$J_z^0(\bar{t}, \bar{z}) = z_c(\bar{t}) + \int_{\bar{t}}^{t_f} R(\sigma)d\sigma = 0 \quad \forall (\bar{t}, \bar{z}) \in C_z. \qquad (5.87)$$

Now, for any given point $(\bar{t}, \bar{z}) \in C_z$, let us consider the initial-value problem (5.82), appearing in the proof of Theorem 5.3.1. Remember that in this initial-value problem, the function $v(t)$ is any given admissible (in the sense of Definition 2.6.1) evader's control. As it was mentioned in the proof of Theorem 5.3.1, the problem (5.82) has the unique solution $z = z^0(t)$, $t \in [\bar{t}, t_f]$. The value $|z^0(t_f)|$ is the SPEGBC outcome $J_z(u_{\text{sg}}^0(\cdot), v(\cdot))$ generated by the pursuer's control $u_{\text{sg}}^0(t, z)$ and the evader's control $v(t)$. Therefore, by virtue of the equation (3.53) (see Subsection 3.5.1) and the equation (5.87), we directly have $0 \le |z^0(t_f)| \le J_z^0(\bar{t}, \bar{z}) = 0$ for all $(\bar{t}, \bar{z}) \in C_z$, i.e., $z^0(t_f) = 0$ for all such points $(\bar{t}, \bar{z})$. This means that $u_{\text{sg}}^0(t, z)$ is a robust capture control from any point $(\bar{t}, \bar{z}) \in C_z$ for the engagement (5.7), (2.52).

To prove that $C_z$ is the capture zone of $u_{\text{sg}}^0(t, z)$, we should show that $u_{\text{sg}}^0(t, z)$ is not a robust capture control from any point $(\bar{t}, \bar{z}) \in S_{[t_0, t_f]} \setminus C_z$ for the engagement (5.7), (2.52). For this purpose, we consider the initial-value problem (5.82) appearing in the proof of Theorem 5.3.1. As it was mentioned in the proof of Theorem 5.3.1, the unique solution $z = z_{\text{sg}}^0(t)$, $t \in [\bar{t}, t_f]$ of (5.82) is the saddle-point trajectory in feedback controls of the SPEGBC, starting at $(\bar{t}, \bar{z})$, and $|z_{\text{sg}}^0(t_f)|$ is the value of the SPEGBC with the starting point $(\bar{t}, \bar{z})$. Since $(\bar{t}, \bar{z}) \notin C_z$, then due to the aforementioned feature of the set $\text{int}(C_z)$ and by virtue of the SPEGBC space decomposition (see Subsection 3.5.3),

$$|z_{\text{sg}}^0(t_f)| > 0. \qquad (5.88)$$

Now, similarly to the proof of Theorem 5.3.1, we consider the initial-value problem (5.85). Remember that in this problem, the function $v_{\text{sg}}^0(t)$, $t \in [\bar{t}, t_f]$ is the time realization of the control $v_{\text{sg}}^0(t, z)$ along the solution $z_{\text{sg}}^0(t)$ of the problem (5.82), and the inequality (5.84) is valid. Due to the proof of Theorem 5.3.1, the unique solution $z_{\text{sg}}^0(t)$ of the problem (5.82) also is the unique solution $z_{\text{sg}}^0(t)$ of the problem (5.85). Thus, for any point $(\bar{t}, \bar{z}) \in S_{[t_0, t_f]} \setminus C_z$, there exists the admissible (in the sense of Definition 2.6.1) evader's control
$v_{sg}^0(t)$, $t \in [\bar{t}, t_f]$ such that the solution $z_{sg}^0(t)$ of the problem (5.85) satisfies the inequality (5.88). This means that $u_{sg}^0(t, z)$ is not a robust capture control from any point $(\bar{t}, \bar{z})$ outside of $C_z$. The latter, along with Remark 30 and the above proven fact that $u_{sg}^0(t, z)$ is a robust capture control from any point $(\bar{t}, \bar{z}) \in C_z$, yields that $C_z$ is the capture zone of $u_{sg}^0(t, z)$, i.e., $C_z = C_z(u_{sg}^0(t, z))$. This completes the proof of the theorem.

### 5.5.4 Proof of Theorem 5.3.5

For any given point $(\bar{t}, \bar{z}) \in C_z$ and admissible (in the sense of Definition 2.6.1) evader's control $v(t)$, $t \in [t_0, t_f]$, we consider the initial-value problem

$$ \frac{dz}{dt} = h_u(t, \phi_{PC})u_{sg,c}(t, z) + h_v(t, \phi_{E0})v(t), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}. \quad (5.89) $$

Using (5.9), (5.12), (5.18), we can rewrite this problem as:

$$ \frac{dz}{dt} = -\frac{a_{P,c}^{\max}(t)}{\tau_P} |\cos(\phi_{PC})| g_P(t) \mathrm{sat}(L(t, z)) + \frac{a_{E,c}^{\max}(t)}{\tau_E} \cos(\phi_{E0}) g_E(t) v(t), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}. \quad (5.90) $$

Note that the problem (5.90) (and, therefore, the problem (5.89)) has the unique continuous solution $z = z_{sg,c}(t)$ in the sense of the particular case of the Krasovskii's constructive motion [9] (see Remark 6).

Comparing the function in the right-hand side of the differential equation in (5.90) with the function $R(t)$ (see the equation (5.11)) and using the inequality $|v(t)| \le 1$, $t \in [t_0, t_f]$, we directly obtain the inequalities

$$ -\frac{a_{P,c}^{\max}(t)}{\tau_P} |\cos(\phi_{PC})| g_P(t) \mathrm{sat}(L(t, z)) + \frac{a_{E,c}^{\max}(t)}{\tau_E} \cos(\phi_{E0}) g_E(t) v(t) \le R(t), $$
where $z \ge z_c(t)$, $t \in (t_{\mathrm{min}}, t_f): t \ne t_{c,i}, i = 1, \dots, l,$

$$ \frac{a_{P,c}^{\max}(t)}{\tau_P} |\cos(\phi_{PC})| g_P(t) \mathrm{sat}(L(t, z)) + \frac{a_{E,c}^{\max}(t)}{\tau_E} \cos(\phi_{E0}) g_E(t) v(t) \ge -R(t), $$
where $z \le -z_c(t)$, $t \in (t_{\mathrm{min}}, t_f): t \ne t_{c,i}, i = 1, \dots, l.$

Due to these inequalities, the solution $z = z_{sg,c}(t)$ of the problem (5.90) (and, therefore, of the problem (5.89)) cannot leave the set $C_z$, i.e., $z_{sg,c}(t) \in C_z$ for all $t \in [\bar{t}, t_f]$. The latter, along with the inclusion $(t_f, 0) \in C_z$, means that $z_{sg,c}(t_f) = 0$. Therefore, $u_{sg,c}(t, z)$ is a robust capture control from any point $(\bar{t}, \bar{z}) \in C_z$ for the engagement (5.7), (2.52).
Now, let us prove that $C_z$ is the capture zone of the control $u_{sg,c}(t, z)$. For this purpose, we consider the following initial-value problem:

$$ \frac{dz}{dt} = h_u(t, \phi_{PC})u_{sg,c}(t, z) + h_v(t, \phi_{E0})v_{sg}^0(t, z), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}, \quad (5.91) $$

where $v_{sg}^0(t, z)$ is of the form (3.74) with $h_v(t) = h_v(t, \phi_{E0})$ and $0 < |\alpha_v(t)| \le 1$. Due to (5.9), (5.12), (5.18), this problem can be rewritten as:

$$ \frac{dz}{dt} = -\frac{a_{P,c}^{\max}(t)}{\tau_P} |\cos(\phi_{PC})| g_P(t) \mathrm{sat}(L(t, z)) + \frac{a_{E,c}^{\max}(t)}{\tau_E} \cos(\phi_{E0}) g_E(t) v_{sg}^0(t, z), \quad t \in [\bar{t}, t_f], \quad z(\bar{t}) = \bar{z}. \quad (5.92) $$

Note that the problem (5.92) (and, therefore, the problem (5.91)) has the unique continuous solution $z = z_{sg,c}^0(t)$ in the sense of the particular case of the Krasovskii's constructive motion [9] (see Remark 20).

Let us observe the following. Due to the definition of $t_{min}$ (see the equation (5.16)), if $t_{min} > t_0$, the function $R(t)$ is positive in a sufficiently small vicinity of $t = t_{min}$ excepting maybe this point itself. Therefore, in this vicinity (excepting maybe $t_{min}$):

$$ \begin{aligned} & -\frac{a_{P,c}^{\max}(t)}{\tau_P} |\cos(\phi_{PC})| g_P(t) \mathrm{sat}(L(t, z)) \\ & + \frac{a_{E,c}^{\max}(t)}{\tau_E} \cos(\phi_{E0}) g_E(t) v_{sg}^0(t, z) \ge R(t), \quad \text{where } z > 0, \\ & \frac{a_{P,c}^{\max}(t)}{\tau_P} |\cos(\phi_{PC})| g_P(t) \mathrm{sat}(L(t, z)) \\ & + \frac{a_{E,c}^{\max}(t)}{\tau_E} \cos(\phi_{E0}) g_E(t) v_{sg}^0(t, z) \le -R(t), \quad \text{where } z < 0. \end{aligned} $$

Using this observation, one can directly conclude that the solution $z = z_{sg,c}^0(t)$ of the problem (5.92) (and, therefore, of the problem (5.91)), starting from the point $(\bar{t}, \bar{z}) \in S_{[t_0, t_f]} \setminus C_z$, $\bar{t} \in [t_0, t_{min})$, satisfies the inequality $|z_{sg,c}^0(t_{min})| > 0$. Note that the latter is valid in both cases, $t_{min} > t_0$ and $t_{min} = t_0$.

Also, let us observe that the control $u_{sg}^0(t, z)$, mentioned in Theorem 5.3.4, satisfies the following condition: $u_{sg}^0(t, z) = u_{sg,c}(t, z)$ for $|z| > z_c(t)$, $t \in (t_{min}, t_f)$. Using this observation and the aforementioned feature of the solution $z = z_{sg,c}^0(t)$ to the problem (5.92) (and, therefore, to the problem (5.91)), we can show quite similarly to the proof of Theorem 5.3.4 that, for any point $(\bar{t}, \bar{z}) \in S_{[t_0, t_f]} \setminus C_z$, there exist the admissible evader's control $v(t)$ such that the solution $z = z_{sg,c}(t)$ of the problem (5.90) (and, therefore, of
the problem (5.89)) satisfies the inequality $|z_{sg,c}(t_f)| > 0$. The latter, along with Remark 30 and the above proven fact that $u_{sg,c}(t, z)$ is a robust capture control from any point $(\bar{t}, \bar{z}) \in C_z$, yields that $C_z$ is the capture zone of $u_{sg,c}(t, z)$, i.e., $C_z = C_z(u_{sg,c}(t, z))$. This completes the proof of the theorem.

### 5.5.5 Proof of Lemma 5.3.10

We start with the proof of the first statement of the lemma, i.e., with the proof of the inequality (5.27). First of all, let us note that the numerator and the denominator in the expression for $H_{uv}(t, \phi_{PC}, \phi_{E0})$ are continuous functions in the interval $[t_{min}, t_f]$. Moreover, these functions are positive for all $t \in [t_{min}, t_f)$, and they equal zero for $t = t_f$. Hence, to prove the validity of (5.27), it is sufficient to show that

$$ \lim_{t \to t_f - 0} H_{uv}(t, \phi_{PC}, \phi_{E0}) > 0. \qquad (5.93) $$

Calculating this limit by application of the L'Hôpitale's rule (see e.g. [1]) and using the equations (5.9) and (5.12), as well as the continuity and positiveness of the functions $a_{P,c}^{\max}(t)$, $a_{E,c}^{\max}(t)$, $t \in [t_0, t_f]$, we obtain

$$ \lim_{t \to t_f - 0} H_{uv}(t, \phi_{PC}, \phi_{E0}) = \frac{(a_{P,c}^{\max}(t_f))^2 \tau_E^2 \cos^2(\phi_{PC})}{(a_{E,c}^{\max}(t_f))^2 \tau_P^2 \cos^2(\phi_{E0})} \lim_{t \to t_f - 0} \frac{g_P(t)}{g_E(t)}. \qquad (5.94) $$

Thus, the calculation of $\lim_{t \to t_f - 0} H_{uv}(t, \phi_{PC}, \phi_{E0})$ is reduced to the calculation of $\lim_{t \to t_f - 0} (g_P(t)/g_E(t))$. Due to the conditions of the lemma, the functions $g_P(t)$ and $g_E(t)$ are twice continuously differentiable in some left-hand vicinity of $t = t_f$ including this point. Hence, the calculation of $\lim_{t \to t_f - 0} (g_P(t)/g_E(t))$ by application twice of the L'Hôpitale's rule and use of the equalities $g_l(t_f) = 0$, $dg_l(t_f)/dt = 0$, $d^2g_l(t_f)/dt^2 = 1$, $(l = P, E)$ (see the proof of Lemma 5.3.3 in Subsection 5.5.2) yield

$$ \lim_{t \to t_f - 0} \frac{g_P(t)}{g_E(t)} = \lim_{t \to t_f - 0} \frac{d^2 g_P(t)/dt^2}{d^2 g_E(t)/dt^2} = 1. $$

The latter, along with the equation (5.94), directly yields the validity of the inequality (5.93), which completes the proof of the first statement of the lemma. The second statement of the lemma directly follows from the equation (5.26), the definition of $\lambda_{inf}(\phi_{PC}, \phi_{E0})$ in (5.27) and the strict form of the inequality (5.28). Thus, the lemma is proven.
### 5.5.6 Proof of Proposition 5.3.12

To prove the proposition, it is sufficiently to show the validity of the inequality

$$ \frac{dH_{uv}(t, \phi_{PC}, \phi_{E0})}{dt} > 0 \quad \forall t \in [t_{\min}, t_f). \qquad (5.95) $$

Using the equation (5.26), we obtain

$$ \frac{dH_{uv}(t, \phi_{PC}, \phi_{E0})}{dt} = \frac{(h_v(t, \phi_{E0}))^2 \int_t^{t_f} (h_u(\sigma, \phi_{PC}))^2 d\sigma - (h_u(t, \phi_{PC}))^2 \int_t^{t_f} (h_v(\sigma, \phi_{E0}))^2 d\sigma}{(\int_t^{t_f} (h_v(\sigma, \phi_{E0}))^2 d\sigma)^2}. \qquad (5.96) $$

Thus, by virtue of (5.95), the function $H_{uv}(t, \phi_{PC}, \phi_{E0})$ is increasing in the interval $[t_{\min}, t_f)$ if the following inequality is valid:

$$ \frac{\int_t^{t_f} (h_u(\sigma, \phi_{PC}))^2 d\sigma}{\int_t^{t_f} (h_v(\sigma, \phi_{E0}))^2 d\sigma} > \frac{(h_u(t, \phi_{PC}))^2}{(h_v(t, \phi_{E0}))^2} \quad \forall t \in [t_{\min}, t_f). \qquad (5.97) $$

Let us treat the expression in the left-hand side of this inequality. By virtue of the Cauchy Theorem (see e.g. [1]), we have

$$ \frac{\int_t^{t_f} (h_u(\sigma, \phi_{PC}))^2 d\sigma}{\int_t^{t_f} (h_v(\sigma, \phi_{E0}))^2 d\sigma} = \frac{(h_u(\tilde{t}, \phi_{PC}))^2}{(h_v(\tilde{t}, \phi_{E0}))^2}, \quad t \in [t_{\min}, t_f), \qquad (5.98) $$

where $\tilde{t} \in (t, t_f)$ is some time-instant depending on $t$.

Since $t < \tilde{t} < t_f$ for all $t \in [t_{\min}, t_f)$ and the function $\tilde{h}_{uv}(t, \phi_{PC}, \phi_{E0})$, given by the equation (5.29), is increasing in the interval $[t_{\min}, t_f)$, then

$$ \frac{(h_u(\tilde{t}, \phi_{PC}))^2}{(h_v(\tilde{t}, \phi_{E0}))^2} > \frac{(h_u(t, \phi_{PC}))^2}{(h_v(t, \phi_{E0}))^2}, \quad t \in [t_{\min}, t_f), \qquad (5.99) $$

which, along with the equation (5.98), yields the validity of the inequality (5.97). This completes the proof of the proposition.

## 5.6 Concluding Remarks and Literature Review

In this chapter, we designed two types of robust capture controls and constructed their capture zones for the engagement considered in Chapter 2.
These design and construction are based on the notion of the *robust capture pursuit* (see Section 2.6) and the feedback saddle-point solution of the zero-sum differential game with linear dynamics and bounded controls (see Chapter 3). Thus, in Section 5.2, the planar pursuit-evasion game was considered. This game is based on the mathematical model of the relative motion between the pursuer and the evader presented in Section 2.5. The cost function of this game is the separation between the players at the final time-instant of the engagement. This separation is called the *miss distance in the engagement*. The objective of the pursuer is to decrease the miss distance as much as possible, while the evader aims to increase the miss distance as much as possible. The pursuit-evasion game of Section 5.2 was analyzed using the results of Chapter 3. Namely, this game was converted to the auxiliary pursuit-evasion game by the proper linear transformation of the state variable. The mathematical model of the engagement between the pursuer and the evader in the auxiliary game is the scalarized mathematical model of the 6-dimensional engagement between the players in the original pursuit-evasion game. Hence the auxiliary game is a scalar game. Particular cases of the planar pursuit-evasion game, studied in Subsection 5.2, were considered in [5, 7, 10, 12].

In Section 5.3, based on the results of Section 5.2, several results about the robust capture pursuit were obtained. Namely, in Theorem 5.3.1 and Corollary 5.3.2, two different kinds of necessary and sufficient conditions for the existence of a robust capture control for the scalarized engagement between the pursuer and the evader are formulated. Sufficient condition for such an existence, directly based on the data of the engagement, is presented in Lemma 5.3.3. It should be noted that Theorem 5.3.1, Corollary 5.3.2 and Lemma 5.3.3 are new assertions, have not been published before. Theorems 5.3.4 and 5.3.5 are devoted to presentation of two types of robust capture controls and their capture zones for the scalarized engagement. Thus, Theorem 5.3.4 presents the robust capture control of the bang-bang form. This control is the pursuer's component of the saddle-point solution to the aforementioned auxiliary scalar pursuit-evasion game and its capture zone is a closed set in the plane (*time*, *scalar state variable*). The interior of this capture zone represents the maximal part of the game's singular region which closure is a simply connected set, and this set contains the point (*final game's time instant*, *zero*). Detailed description of the game's space decomposition, including the definitions of the singular and regular regions and their construction, is presented in Section 3.5. Theorem 5.3.5 presents the continuous with respect to the state variable robust capture control and its capture zone for the scalarized engagement. This control is the saturation of the properly defined odd function with respect to the state variable. The capture zone of
this control coincides with the one of the robust capture control presented in Theorem 5.3.4. Various particular cases of the results, stated in Theorems 5.3.4 and 5.3.5, can be found in [3, 5, 6, 10, 12, 13], while Theorems 5.3.4 and 5.3.5 themselves are new ones, published for the first time in the literature in the present chapter.

Theorem 5.3.7 establishes the connection between robust capture controls for the scalarized engagement and for the original 6-dimensional engagement. Based on the robust capture controls and the corresponding capture zones, mentioned in Theorems 5.3.4 and 5.3.5, Corollaries 5.3.8 and 5.3.9 present the robust capture controls and the corresponding capture zones for the original 6-dimensional engagement between the pursuer and the evader.

Two types of the odd function, participating in the aforementioned robust capture control, were constructed in Subsections 5.3.1 and 5.3.2, respectively. The first type of this function, considered in Subsection 5.3.1, is based on the equation of the upper boundary of the corresponding capture zone. Particular case of this type can be found in [5, 6, 13]. The second type of this function, considered in Subsection 5.3.2, is based on the feedback solution to the scalar zero-sum linear-quadratic differential game with cheap controls (the game itself is studied in Chapter 4). Lemma 5.3.10, Corollary 5.3.11 and Proposition 5.3.12 present various sufficient conditions, providing the feasibility of the aforementioned function of the second type. Particular cases of this type can be found in [13, 14]. Lemma 5.3.10 is the extension of the result of [14] to the case where the participants of the engagement have any aspect angles in the initial collision course (the notion of these aspect angles is presented in Section 2.3). Corollary 5.3.11 and Proposition 5.3.12 are new assertions, have not been published before.

In Section 5.4, two examples of the planar engagement scenario between two flying vehicles were presented (see Subsections 5.4.1 and 5.4.2). In each example, the decomposition of the space of the scalar pursuit-evasion game is presented. The robust capture controls and the corresponding capture zones for the scalarized engagement, mentioned in Theorems 5.3.4 and 5.3.5, are derived. Based on these results, the capture controls and the corresponding capture zones for the original 6-dimensional engagement, mentioned in Corollaries 5.3.8 and 5.3.9, are obtained. It should be noted that in Subsection 5.4.1, it was considered the case where the determining function changes its sign twice during the engagement. Such a case has not yet been considered in the literature. Definition of the determining function and its use in the game's space decomposition are presented in Section 3.5. In Subsection 5.4.3, the robust capture controls derived in Subsections 5.4.1 and 5.4.2 are evaluated by numerical simulations for the nonlinear engagement model and subject to the presence of a random wind disturbance. The simulations re-
sults show that, in spite of the model nonlinearity and the presence of the
wind, all the controls yield very small miss distances. Namely, the largest
miss distance yielding by the controls of Subsection 5.4.1 is less than 30 [cm],
while the largest miss distance yielding by the controls of Subsection 5.4.2
is less than 20 [cm]. These simulations results, along with the undesirable
chattering yielding by the bang-bang robust capture control, clearly show
the preference of using the aforementioned continuous (with respect to the
state variables) robust capture controls in real-life implementations.

In Section 5.5, rigorous proofs of Theorems 5.3.1-5.3.5, Lemmas 5.3.3-5.3.10 and Proposition 5.3.12 were done.
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