Approximation procedures for an abstract LQ-optimal control problem (Q2711387)

This paper deals with the approximation of solutions of an LQ-optimal control problem stated in an abstract form. The input-state relationship is written as \(x = Tu\) (in a typical application \(T\) is the mapping from the input function \(u\) to the state trajectory \(x\), given maybe by a differential or integral equation). Here \(x \in X\) and \(u \in U \subset Y\), where \(X\) and \(Y\) are separable Hilbert spaces and \(U\) is a closed and convex subset of \(Y\). The cost functional is written in the form \(C(x,u) = \langle Px,x \rangle + \langle Qu,u \rangle\), where \(P\) and \(Q\) are strictly positive operators on \(X\), respectively \(Y\). The first approximation method is of Ritz-Galerkin type: the space \(Y\) is mapped onto \(l^2\) by means of an orthonormal basis, \(u\) is approximated by a truncated Fourier series, and a minimizing sequence \(u_n\) is constructed (each \(u_n\) has a finite Fourier series, and the cost tends to the optimal cost as \(n \to \infty\), but the sequence \(u_n\) itself may or may not converge to a minimizing \(u\)). Special attention is paid to the influence of the closed and convex set \(U\) to which \(u\) is restricted. In one example \(U\) is the unit ball in \(Y\) or an ellipsoid, and in the next example the system is a standard linear discrete time system in \(l^2\). In these two examples \(U\) is approximated by finite-dimensional ellipsoids in construction of the minimizing sequence \(u_n\). In the case of a general closed and convex \(U\) more complicated approximations of \(U\) in the form of convex polyhedrals are needed. The final example is an application to an abstract Volterra equation.