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Regularity of the minimum time function and minimum energy problems: The linear case. (English) Zbl 0958.49014

This substantial, well-written paper is devoted to the study of continuity properties of the minimum time function associated with a linear dynamical system, when the state space is a separable Hilbert space \(H\) and the set of admissible controllers is \(\mathcal{U}_{ad}=\{ u\in L^p(0,\infty ;U):\left|u\right|_p\leq \rho\} ,\) \(p\in ( 1,\infty)\), with \(U\) being a separable Hilbert space, too. The key tool in the investigation of various continuity properties of the Bellman function \(T_p\) is the minimum energy function. New results interesting in itself are proven about the regularity of the minimal energy. It is shown that, under reasonable hypotheses, for \(p\)-null controllable systems, the minimum time is implicitly defined by the minimum energy. Using this fact, it is proven that the growth rate of the minimum time function near to the origin is connected to the explosion rate of the minimum energy as the time horizon tends to zero. As a consequence of this connection, the local uniform continuity together with the modulus of continuity is derived. As a particular case, the local Hölder continuity with a certain exponent is obtained. Authors prove that results concerning some topological properties of the reachable set known in the case \(p=\infty \) can be generalized to the case \(p\in (1,\infty).\) Part of the results are new also in the finite dimensional case.

MSC:

49K27 Optimality conditions for problems in abstract spaces
47A75 Eigenvalue problems for linear operators
93C20 Control/observation systems governed by partial differential equations
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