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Dynamical Systems with Unbounded Time Interval in Engineering
Necessary Conditions and Sufficient Conditions for Optimality
Asymptotic Stability and the Turnpike Property in Some Simple Con
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absolutely continuous admissible control admissible pair admissible trajectories assume Assumption 4.1 asymptotic stability Bellman equation bounded calculus of variations Chapter concave function consider constant constraint continuous function control system convex convex set cost defined denote deterministic differential equation discount rate emanating from Xo Example exists an overtaking extremal trajectory G-supported trajectories given growth condition Hamilton-Jacobi equation Hamiltonian system Hence holds horizon optimal control implies infinite horizon optimal infinity initial value integral IRn x IRn Lebesgue measure Lemma lim sup liminf linear Lipschitz continuous matrix maximal maximum principle measurable function minimal modified Hamiltonian system Moreover negative definite nonpositive observe obtained optimal control problem overtaking optimal solution pqxu problem of Lagrange proof of Theorem Proposition reduction to finite Remark Riccati equation Section semigroup sequence stationary point strongly optimal solution sufficient conditions trajectory emanating turnpike property uniformly upper semicontinuous variable weakly overtaking optimal