## 2 Problem Description  
Consider the continuous-time nonlinear system with external disturbance as  
\[\dot{x}=f(x(t))+g(x(t))u(t)+h(x(t))v(t) \tag{1}\]  
where \(x\in\mathbb{R}^{n}\) denotes the states vector of system. \(u(t)=\{(u_{1},u_{2},\cdots,u_{m})\in\mathbb{R}^{m}:\|u_{i}\|\leq u_{M},i=1,2, \cdots,m\}\), \(u_{M}\) is a positive upper bound. \(f(x)\in\mathbb{R}^{n}\) is drift dynamic, and has \(f(0)=0\). \(g(x)\in\mathbb{R}^{n\times m}\) is input coupling dynamic. \(h(x)\in\mathbb{R}^{n\times q}\) and \(v(t)\in\mathbb{R}^{q}\) are the disturbance dynamics and bounded external disturbance. \(v_{M}\) is a positive constant with \(\|v(t)\|\leq v_{M}\). Since the system is affected by constrained input, \(M(u)\) is defined as [39]  
\[ \begin{split} M(u)&=2\int_{0}^{u}u_{M}\tanh^{-\mathsf{ T}}(\mu/u_{M})Rd\mu\\ &=2\sum_{i=1}^{m}\int_{0}^{u_{i}}u_{M}\tanh^{-\mathsf{T}}(\mu_{i}/u_{M })Rd\mu_{i}\end{split}  \tag{2}\]  
where \(M(u)\) represents nonquadratic function, and \(R\) is assumed to be diagonal matrix. \(\tanh^{-\mathsf{T}}(\cdot)\) is a hyperbolic tangent function, which is treated as the constrained input. For the \(H_{\infty}\) control problem with constrained input, we need to reduce the impact of external disturbances on system performance. For any \(v(t)\in L_{2}[0,\infty)\), the closed-loop system (1) exists  
\[\int_{t}^{\infty}\Bigl{(}x^{\mathsf{T}}Qx+M(u)\Bigr{)}d\tau\leq\gamma^{2}\int _{t}^{\infty}v^{\mathsf{T}}(\tau)v(\tau)d\tau \tag{3}\]  
and this is, \(L_{2}\)-gain not larger than \(\gamma\) is satisfied, where \(Q\in\mathbb{R}^{n\times n}\) is a symmetric positive matrix. \(\gamma>0\) denotes the level of disturbance attenuation. The design intention of the \(H_{\infty}\) optimal control problem is to find a control law that not only guarantees the asymptotic stability of the system (1) but also the disturbance attenuation condition (3) holds. Then, we need to define the following value function  
\[J(x(t),u,v)=\int_{t}^{\infty}U\bigl{(}x(\tau),u(\tau),v(\tau)\bigr{)}d\tau \tag{4}\]  
where \(U(x,u,v)=x^{\mathsf{T}}Qx+M(u)-\gamma^{2}v^{\mathsf{T}}v\). To achieve the above goals, the \(H_{\infty}\) control problem is treated as the zero-sum game problem. By the minimax optimization principle, the perturbation policy is used as a decision-maker and maximizes its value, while the control policy acts as another decision-maker and minimizes its value. The following optimal value function is indicated as  
\[V^{*}(x)=J(x(t),u^{*},v^{*})=\min_{u}\max_{v}\int_{t}^{\infty}U\bigl{(}x(\tau),u(\tau),v(\tau)\bigr{)}d\tau \tag{5}\]  
Assume that the \(V(x)=J(x,u,v)\) is continuous and differentiable, and the Bellman equation can be given as  
\[\
abla V^{\mathsf{T}}(f+gu+hv)+x^{\mathsf{T}}Qx+M(u)-\gamma^{2}v^{\mathsf{T} }v=0 \tag{6}\]  
Then, the Hamiltonian function is defined as  
\[H(x,\
abla V,u,v)=\
abla{V^{*}}^{\mathsf{T}}(f+gu+hv)+M(u)+x^{\mathsf{T}}Qx- \gamma^{2}v^{\mathsf{T}}v \tag{7}\]  
**Definition 1** ([12]). : _A control law is defined as admission with respect to the value function (4) on a compact set \(\Lambda\). If \(u(0)=0\) and \(v(0)=0\), \(u(x)\) can ensure system (1) stability on \(\Lambda\) and for any \(x_{0}\in\Lambda\) the value funcion \(V(x_{0})\) is finite._  
By stationarity conditions \(\partial H/\partial u=0\) and \(\partial H/\partial v=0\), the optimal control law \(u^{*}(x)\) and the disturbance law \(v^{*}(x)\) are expressed as  
\[u^{*}(x)=-u_{M}\tanh\left(\frac{1}{2u_{M}}R^{-1}g^{\mathsf{T}}(x)\
abla V^{*}( x)\right) \tag{8}\]  
\[v^{*}(x)=\frac{1}{2}\gamma^{-2}h^{\mathsf{T}}(x)\
abla V^{*}(x) \tag{9}\]  
Submitting (8) into (2), \(M(u^{*})\) can be obtained  
\[M(u^{*})=u_{M}\
abla{V^{*}}^{\mathsf{T}}g\tanh(\Psi)+u_{M}^{2}\bar{R}\ln\left( \frac{1}{2}-\tanh^{2}(\Psi)\right) \tag{10}\]where \(\Psi=1/(2u_{M})R^{-1}g^{\mathsf{T}}\
abla V^{*}(x)\), and \(\underline{1}\) indicates the elements are all 1 column vectors, \(\bar{R}\) is a row vector made up of the diagonal elements of \(R\).