The pursuer set in two-point optimization problems (Q2388060)

Let \(\tilde{u}(\cdot)\in V(\Delta),\Delta=[0,T]\), be an optimal control for the problem \[ \dot{x}=f(x,u),\;x\in R^n \] \[ x(0)=x_0,\;x(T)=x_1 \] \[ \int_{0}^{T}g(x(t),u(t))dt \rightarrow \min . \] According to the Pontryagin maximum principle there exists a nonzero solution \(\Psi(t)=(\psi_0,\psi(t))\) \(\in R^{1+n},\psi_0=\text{const.}\leq 0\), of the equation \[ \dot{\psi}=-\frac{\partial H(\Psi,\tilde{x}(t),\tilde{u}(t))}{\partial x} \] such that the maximum condition \[ H(\Psi,\tilde{x}(t),\tilde{u}(t))=\max_{u\in U}H(\Psi,\tilde{x},u) \] is fulfilled. Here \(V(U)\) is the set of measurable functions \( u(t)\in U,\;U \subset R^r\) -- compact set; \(H=\psi_0 g+\langle \psi,f\rangle.\) Without loss of generality one can assume that the vector \(\varphi=\Psi(T)\) satisfies the condition \(| \varphi | =1.\) By \(\Pi(\tilde{u}(\cdot))\) we denote the set of all normalized vectors \(\varphi\) corresponding to the optimal control \(\tilde{u}(\cdot).\) The set \(\Pi(\tilde{u}(\cdot))\) is called the pursuer set. The paper deals with the geometric description of the set \(\Pi(\tilde{u}(\cdot)).\) The following results are obtained: Theorem 1. The set \(\Pi(\tilde{u}(\cdot))\) consists of unit support vectors \(b=(b_0,b_1,\dots,b_n)\) of the convex set \[ D(T)=\bigg\{ \int_{0}^{T}\Phi(T,s)[F(\tilde{x}(t),u(t))-F(\tilde{x}(t),\tilde{u}(t))]ds : u(\cdot)\in V(U)\bigg \} \] at the point \(0\in R^{1+n}\) satisfying the inequality \( b_0\leq 0.\) Here \(\Phi(t,s),s\in [0,t]\), is the matrix function satisfying the equation \[ \frac{\partial \Phi(t,s)}{\partial s}=F_{x}(\tilde{x}(s),\tilde{u}(s))\Phi(t,s) \] and the condition \( \Psi(t,t)=E ;\; F=(g,f)^{*}.\) Theorem 2. The set \[ \bigcup_{\lambda\in [0,1]}{\lambda \Pi(\tilde{u}(\cdot)) } \] is nonempty convex compact. Theorem 3. A unit vector \( b \) is a support vector of the convex set \(D(T)\) at the point \(0\in R^{1+n}\) if and only if \[ \int_{0}^{T}\max_{a\in \Omega_{1}(s)}\langle a,b\rangle ds=0, \] where \( \Omega_{1}(s)=\Phi(T,s)\Omega(s)\), \(\Omega(s)=F(\tilde{x}(s),U)-F(\tilde{x}(s),\tilde{u}(s)). \)