Generalized optimal control of linear systems with distributed parameters
The book represents an attempt to give a unified approach for a special kind of optimal control problems governed by linear partial differential equations with impulse-like controls in the right-hand side. As a characteristic prototype one can consider the minimization of the functional \[ I(u):= \int_Q a(x,t)(u(x, t)- g(x,t))^2 dx dt \] over solutions \(u\) of the parabolic equation \[ \begin{multlined} (Lu)(x,t):= u_t(x,t)- \sum^n_{i,j=1} {\partial\over\partial x_i} (a_{ij}(x) u_{x_j})\\ = f(x,t)+ \sum^N_{s=1} \delta(t- \tau_s) \varphi_s(x)+ \sum^N_{k=1} \delta(x_1- \xi_k) \psi_k(x_2,\dots, x_n, t)\\ := (f+ F(h))(x, t),\qquad (x,t)\in Q,\end{multlined}\tag{1} \] \[ u|_{t=0}= 0,\qquad u|_{\partial\Omega}= 0, \] when the control \(h:= \{\tau_1,\dots, \tau_N; \varphi_1,\dots, \varphi_N; \xi_1,\dots, \xi_M;\psi_1,\dots, \psi_M\}\) runs over a given set \(U_0\) from an appropriate space \(U\). Here \(Q= \Omega\times (0, T)\) is a cylinder with \(\Omega\subset \mathbb{R}^n\) being a bounded set with a smooth boundary \(\partial\Omega\) and \(\delta\) is the Dirac function. Because the right-hand side of (1) does not belong to Lebesgue spaces one has to consider generalized solutions of (1). This problem in the book is solved by reformulating (1) as an equation in appropriate Hilbert spaces: find \(u\in H_1\) such that \[ \langle L^+v,u\rangle_{H_1}= \langle f+F(h),v\rangle_{H_2} \forall v\in H_2: L^+v\in H^-_1,\tag{2} \] where \(L^+\) is the algebraically adjoint to the operator \(L\) and by \(H^-\) is denoted the dual space to a Hilbert space \(H\). The most important part of the book are various, more or less standard, estimates, which ensure that the equation (2) is uniquely solvable and that the mapping \(L^{-1}\), \(L^{-1}(f+ F(h))= u\), is continuous from \(H^-_2\) to \(H_1\). That is done for specific types of operators \(L\) of the kind: parabolic (as in (1)), hyperbolic, pseudoparabolic, pseudohyperbolic and Sobolev type operators. The essence here is the right account of the interplay between properties of the operator \(L\) and the singularities of \(F(h)\). The knowledge of continuity of \(L^{-1}: H^-_2\to H_1\) is exploited to obtain existence theorems for optimal control problems (simple conditions there are weak continuity of the mapping \(F: U\to H^-_2\) and weak lower semicontinuity of the functional \(I: H_1\to R\)) and differentiability properties of the cost functional \(I\) with respect to the control \(h\). It is necessary to mention here that the differentiability of \(I\) is considered only under the existence of the Fréchet derivative of the mapping \(h\to F(h)\in H^-_2\) and as such it fails for most cases if the parameters \(\{\tau_s\}\) are considered. To overcome this difficulty, a regularization (by standard mollifiers) of the right-hand side of equations is proposed and corresponding convergence results are given. Shortages of the book are the lack of an introduction, the lack of simple examples, which illustrate what the general results give for concrete cases, and a lot of grammatical (sometimes even linguistic) errors.
scientific article; zbMATH DE number 1840598