On the convergence of gradient methods in solving singular optimal control problems (Q2739956)

Let the system state \(u(h)\) be a solution of the linear differential equation \(Lu=f(h)\), where \(L:L_2\to L_2\), \(L_2=L_2(Q)\) is an operator with domain of definition \(D(L)\) which is dense in \(L_2\); \(h\in U_{\partial}\) is a control; \(f(h):H\to W_{*}^{-}\); \(H\) is a space of controls; \(U_{\partial}\subset H\) is the closed, convex and bounded set of admissible controls. It is necessary to minimize the quality criterion \(J(h)= \int_{Q}(u(h)-u_{\partial})^2 dQ\), where \(u_{\partial}\in L_2\) is a given state of the system. The author proves the following theorem.NEWLINENEWLINENEWLINELet \(f\) be Fréchet differentiable with Hölder-Lipschitz continuous derivative. Then if the functional \(J(h)\) has at most countable number of values on the set NEWLINE\[NEWLINEH^{*}=\{h^{*}\in U_{\partial}:\inf_{h\in U_{\partial}} (J'(h^{*}),h-h^{*})\geq 0\},NEWLINE\]NEWLINE then the limit of an arbitrary convergent subsequence \(h^{s_{k}}\) belongs to \(H^{*}\). Here \(h^{s+1}= h^{s}+\rho_{s}({\bar h}^{s}-h^{s})\), \(0<\rho_{s}<1\), \(\lim_{s\to\infty}\rho_{s}=0\), \(\sum_{s=0}^{\infty} \rho_{s}=+\infty\).