×

Approximation methods in problems of optimization and control. (Metody approksimatsij v zadachakh optimizatsii i upravleniya). (Russian) Zbl 0643.49001

Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 360 p. R. 5.00 (1988).
After a detailed account of nonsmooth analysis, necessary conditions for an extremum in nonsmooth optimization are derived from theorems on approximation and separation of sets. Nonsmooth optimal control problems (including phase constraints) are approached by approximating the problem, relaxing the constraints and the objective function, leading to a maximum principle and generalizations. Piecewise approximations of trajectories are also discussed, leading to an approximate Hamiltonian, and an approximate maximum principle (maximum within \(\epsilon)\). Stability conditions are obtained for the convergence of a sequence of Hamiltonians obtained from discrete approximations. Some existence results follow for optimal controls.
Reviewer: B.Craven

MSC:

49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49J52 Nonsmooth analysis
49M05 Numerical methods based on necessary conditions
49K15 Optimality conditions for problems involving ordinary differential equations
49K40 Sensitivity, stability, well-posedness
90C30 Nonlinear programming