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If we substitute ϕ for x and F for u in (2,3,8) we find that P (t) := P (t1 − t) satisfies the Riccati differential equation (1,6,5) and the value function V is continuously differentiable. Neubrander,Vector-valued Laplace Transforms and Cauchy Problems, Birkh¨auser 2001. Press 1995. Falb, Optimal Control, Springer 1975. 31-78 in Lecture Notes in Control and Information Sciences 135, Springer 1989. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkh¨auser 1997.

58. The proof of Theorem 1,57 depends essentialy on boundedness of both operators B and C (cf. Example 1,55). The infinite horizon case for a bounded operator C is fully discussed in [L-T2]. PartIII and [L-Y]. The case of an unbounded operator C is discussed in [D-F-M]. Weiss. The interested reader can consult their webb pages. e. X = Rn , U = Rm , Y = Rp . Including of nonlinear partial differential equations would require many prerequisities. g. [L-Y]. 1 CONTROLLABILITY FOR LINEAR SYSTEMS WITH CONSTRAINTS Assume that we need to control an actual linear systems given by (1,2,1) from an initial state x0 to another state x1 in given time T and these two points are distant from each other.

For another proof the reader can consult [L-M]. If x0 ∈ C then x0 ∈ CT for some time T > 0 and hence also x0 ∈ Cτ for all τ > T. It follows from w∗ −compactness of Mt and w∗ −continuity of Lt that Ct = Ct0 . t>t0 This means that there exists a minimal time tˆ for which x0 ∈ Ctˆ. A corresponding control uˆ which steers x0 into 0 at the minimal time tˆ can be characterized by the following special form of the maximum principle (see Section 2,3). 4. Let M be a convex compact neighborhood of the origin.

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