On the optimal control of systems described by implicit evolution equations

Abstract

We obtain conditions for the existence and uniqueness of an optimal control for the linear nonstationary operator-differential equation $$ \frac{d} {{dt}}[A(t)y(t)] + B(t)y(t) = K(t)u(t) + f(t) $$ with a quadratic performance criterion. The operators A(t) and B(t) are closed and may have nontrivial kernels. The results are applied to differential-algebraic equations and to partial differential equations that do not belong to the Cauchy-Kowalewskaya type.