Overlapping quadratic optimal control of linear time-varying commutative systems.

*(English)*Zbl 1012.93011The so-called strategy of “generalized selection of complementary matrices” for overlapping state linear quadratic optimal control from linear time-invariant systems is extended to linear time-varying systems satisfying a certain commutative property. More precisely, given an optimal control problem associated with the systems
\[
\dot x(t)= A(t)x(t)+ B(t)u(t),\tag{1}
\]
\[
\dot{\widetilde x}(t)=\widetilde A(t)\widetilde x(t)+\widetilde B(t)\widetilde u(t)\tag{2}
\]
and assuming that the dimensions of the state and input vectors \(x(t)\) and \(u(t)\) of (1) are smaller than (or at most equal) to those of \(x(t)\) and \(u(t)\) in (2) so that the “inclusion principle” holds, explicit conditions on the “complementary matrices” are displayed by including a systematic computational procedure for their selection. In this respect, the vectors and matrices in (1), (2) are related by
\[
\widetilde x(t)= Vx(t);\quad x=U\widetilde x(t);\quad\widetilde u(t)= Ru(t);\quad u(t)= Q\widetilde u(t)
\]
(\(V\), \(U\), \(R\) and \(Q\) constant matrices)
\[
\widetilde A(t)= VA(t) U+ M(t);\quad \widetilde B(t)= VB(t) Q+ N(t).
\]
Here \(M(t)\) and \(N(t)\) are the “complementary matrices” whose appropriate selection turns out to be the key point in this kind of problem. This is the main contribution of the paper, as worked out in Section 3. An illustrative numerical example is also presented in Section 4.

Reviewer: Pablo Gonzalez-Vera (La Laguna)