H. O. Fattorini

Time and norm optimal controls: A survey of recent results and open problems

We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagins maximum principle. In particular, we discuss optimality (or nonoptimality) of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.

Time optimality and the maximum principle in infinite dimension

We present three counterexamples related to the maximum principle for the time optimal problem. The system is linear, infinite dimensional, with point target and “full” control

Linear Control Systems in Sequence Spaces

Pontryagin’s maximum principle in its infinite-dimensional version provides (separate) necessary and sufficient conditions for both time and norm optimality for the system y′ = Ay + u (A an infinitesimal generator). The question whether targets in D(A) guarantee a smooth costate has been open. We show the answer is “no” by means of a counterexample involving an analytic semigroup. Another analytic semigroup sheds some light on other subjects such as the existence of hypersingular time optimal controls (thus answering another open question) and the characterization of the reachable space and of singular functionals in its dual.

Time and Norm Optimality of Weakly Singular Controls

Let \( \bar{u}(t) \)be a control that satisfies the infinite-dimensional version of Pontryagin’s maximum principle for a linear control system, and let \( {z}(t) \)be the costate associated with \( \bar{u}(t) \). It is known that integrability of \( {z}(t) \)in the control interval [0, T] guarantees that \( \bar{u}(t) \)is time and norm optimal. However, there are examples where optimality holds (or does not hold) when \( {z}(t) \)is not integrable. This paper presents examples of both cases for a particular semigroup (the right translation semigroup in \( {L^2}(0,\infty )\)).

Smoothness of the costate and the target in the time and norm optimal problems

We show that, in the setting of general semigroups there is in general no relation between smoothness of the target and smoothness of the costate for time and norm optimal controls. However, there exist relations in particular cases, such as self adjoint and unitary semigroups. †Dedicated to Prof. N.U. Ahmed on the occasion of his 70th birthday.

Time and Norm Optimal Controls for Linear Parabolic Equations: Necessary and Sufficient Conditions

We prove a sufficient condition for time and norm optimality of controls for linear parabolic distributed parameter systems with a pointwise bound on the controls, and explore its interplay with existing necessary conditions. This sufficient condition produces simple examples of optimal controls.