## 2 Statement of the Problem  
We consider the nonlinear continuous dynamical system described by the vector differential equation:  
\[\dot{x}(t)=f(t,x(t),u(t)), \tag{1}\]  
where \(f(t,x,u)\) is a given continuous function, \(t\in T=[t_{0};t_{1}]\) is a continuous time and the initial moment \(t_{0}\) and final moment \(t_{1}\) are specified; \(x\in R^{n}\) is a state vector; \(u\in U\subseteq R^{q}\) is a control vector and \(U=[a_{1},b_{1}]\times\ldots\times[a_{q},b_{q}]\) is a set of allowable values of control.  
The initial conditions are specified as:  
\[x(t_{0})=x_{0}\in\Omega\subset R^{n}, \tag{2}\]  
where \(\Omega\) is a set with positive measure (mes \(\Omega>0\)) and a piecewise smooth boundary. It characterizes the uncertainty in setting the initial conditions.  
The model of the measuring system is described by the relation:  
\[z(t)=h(t,x(t)), \tag{3}\]  
where \(z\in R^{m}\) is an output vector and \(h(t,x)\) is a given continuous function. The information coming from the model of the measuring system arrives at the input of the state observer, producing an estimate of the state vector.  
We suppose that it is possible to obtain an estimate of the state vector using a nonlinear observer of the form:  
\[\frac{d\hat{x}(t)}{dt}=f(t,\hat{x}(t),u(t,\hat{x}(t)))+K(t,\hat{x}(t))[z(t)-h(t,\hat{x}(t))], \tag{4}\]  
\[\hat{x}(t_{0})=\hat{x}_{0}, \tag{5}\]  
where \(\hat{x}(t)\) is a state vector estimate, \(\hat{x}_{0}\in\Omega\) is an initial estimate and \(K(t,\hat{x})\in R^{n\times m}\) is an unknown continuous \(n\times m\) matrix function. This matrix is considered as a feedback control of the observation process. The state vector estimate is used also in the plant control law \(u(t,\hat{x})\).  
We define the set of admissible control laws \(\mathbf{U}\) by functions \((u(t,\hat{x}),K(t,\hat{x}))\), where \(\forall t\in T\), the plant control \(u(t)=u(t,\hat{x}(t))\in U\) is a piecewise continuous and the observer control \(K(t)=K(t,\hat{x}(t))\in R^{n\times m}\) is a continuous function. It is assumed that the solution of the system of Equations (1) and (4) with the initial conditions (2), (5) taking into account (3), exists and is unique.  
The performance index for a separate trajectory:  
\[I(x_{0},u(t,\hat{x}(t)),K(t,\hat{x}(t)))=\int\limits_{t_{0}}^{t_{1}}f^{0}(t,x(t ),u(t,\hat{x}(t)),K(t,\hat{x}(t)))dt+F(x(t_{1})), \tag{6}\]  
where \(f^{0}(t,x,u,K),F(x)\) are given continuous functions.  
We associate the pencil of trajectories of the system of Equations (1) and (4) with each admissible control law \((u(t,\hat{x}),K(t,\hat{x}))\in\mathbf{U}\) and the set \(\Omega\) of initial states:  
\[X(t,u(t,\hat{x}),K(t,\hat{x}))=\cup\{x(t,u(t,\hat{x}),K(t,\hat{x}),x_{0}),\hat {x}(t,\mathrm{U}(t,\hat{x}),K(t,\hat{x}),\hat{x}_{0})|x_{0}\in\Omega\},\]  
that is, the union of the system of Equations (1) and (4) and solutions for all possible initial states from the set \(\Omega\).  
The performance index for the pencil of trajectories control to be minimized is:  
\[J[\mathrm{U}(t,\hat{x}),K(t,\hat{x})]=\int\limits_{\Omega}I(x_{0},\mathrm{U}(t,\hat{x}(t)),K(t,\hat{x}(t)))dx_{0}/\mathrm{mes}\,\Omega\,. \tag{7}\]  
The optimal control problem is to choose the control policy \((u*(t,\hat{x}),K*(t,\hat{x}))\in\mathbf{U}\) so that performance index (7) is minimized:  
\[J[\mathrm{U}^{*}(t,\hat{x}),K^{*}(t,\hat{x})]=\min\limits_{(u(t,\hat{x}),K(t, \hat{x}))\in u}J[u(t,\hat{x}),K(t,\hat{x})] \tag{8}\]  
Since the average value of performance index (6) is minimized on the set of initial states \(\Omega\), the required control is called optimal on average.