Minimum Time and Minimum Energy for Linear Systems; a Variational Approach
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Minimum Time and Minimum Energy for Linear Systems; a Variational Approach O. Cârja˘ 1 · A. I. Lazu2
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we develop a variational approach for the norm optimal control problem in an abstract general setting for linear systems. This technique is also related to the classical pseudoinverse for linear and bounded operators between Hilbert spaces. Characterizations for time optimal controls are also given. Keywords Time optimal control · Norm optimal control · Variational approach · Boundary control systems Mathematics Subject Classification 93C25 · 93C20
1 Introduction Let X and U be two Banach spaces and consider the control system represented by y (t, x, u) = S (t) x + V (t) u, t > 0, (1) y (0, x, u) = x, where y is the state, t the time and u the control. Here, {S(t); t ≥ 0} is a C0 -semigroup on X and V (t) , t > 0, is a family of bounded linear operators, V (t) : L ∞ (0, t; U ) → X , such that the following composition property is satisfied V (t1 + t2 ) u = S (t2 ) V (t1 ) u + V (t2 ) Jt1 u,
B
(2)
O. Cârj˘a [email protected] A. I. Lazu [email protected]
1
Department of Mathematics, “Al. I. Cuza” University, Ia¸si 700506 and “Octav Mayer” Mathematics Institute, Romanian Academy, 700505 Iasi, Romania
2
Department of Mathematics, “Gh. Asachi” Technical University, 700506 Iasi, Romania
123
Applied Mathematics & Optimization
for all t1 ,t2 > 0 and u ∈ L ∞ (0, t1 + t2 ; U ), where Jt1 is a translation operator given by Jt1 u (s) = u (s + t1 ) for s ≥ 0. Clearly, in V (t1 ) u we have considered the restriction of u to [0, t1 ]. Further, assume that for each u ∈ L ∞ (0, +∞; U ) the function t → V (t)u is continuous and limt→0 V (t)u = 0. Equation (1) may be seen as a variation of constants formula associated with an evolution equation. The typical example is the distributed control system y (t) = Ay(t) + Bu(t), y(0) = x, where A is the generator of {S(t); t ≥ 0} and B : U → X is linear and bounded. In this case, V (t)u =
t
S(t − s)Bu(s)ds.
(3)
0
The operator B could be also unbounded. This case covers the boundary control problems. See [23, p. 78] for a representation formula of V (t). See also the example presented in the last section. We assume that X is reflexive, U is a Hilbert space, but it could be also reflexive (see Remark 11), and the system (1) is null controllable at any time t > 0. We also assume that for every t > 0, V (t) = H (t)∗ , for some H (t) : X ∗ → L 1 (0, t; U ). For V (t) given by (3) we have H (t)x ∗ = B ∗ S ∗ (t − ·)x ∗ . H (t) is the observability operator, which means H (t)x ∗ = B ∗ p, where p is the solution of the adjoint equation p (s) = −A∗ p(s), p(t) = x ∗ . In this paper we consider the following two optimal control problems associated to (1). (NO) Norm optimal control problem. For T > 0 fixed, minimize the norm of controls u by which the initial point x can be steered to zero in time T , i.e., satisfying the equation V (T )u = S(T )x. (TO) Time optimal control problem. F
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