Bolza Problems with General Time Constraints

This work provides necessary conditions for optimality in problems of optimal control expressed as instances of the generalized problem of Bolza, with the added feature that the fundamental planning interval is allowed to vary. A central product of the analysis is a generalization of the conservation-of-Hamiltonian condition for problems on either fixed or variable intervals. The results, which allow for unprecedented generality in the problem data, are derived from known properties of fixed-interval problems under the hypothesis that the time-dependence of the objective integrand has the same modest level of regularity as the state-dependence.

[1]  Harry Zheng,et al.  The Extended Euler--Lagrange Condition for Nonconvex Variational Problems , 1997 .

[2]  Giuseppe Buttazzo,et al.  Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands , 1989 .

[3]  Sergei M. Aseev,et al.  Investigation of the Degeneracy Phenomenon of the Maximum Principle for Optimal Control Problems with State Constraints , 1997 .

[4]  Philip D. Loewen,et al.  An intermediate existence theory in the calculus of variations , 1989 .

[5]  Philip D. Loewen,et al.  Differential inclusions with free time , 1988 .

[6]  Philip D. Loewen,et al.  Optimal Control of Unbounded Differential Inclusions , 1994, SIAM Journal on Control and Optimization.

[7]  Jean-Pierre Aubin,et al.  Lipschitz Behavior of Solutions to Convex Minimization Problems , 1984, Math. Oper. Res..

[8]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[9]  Existence and regularity in the small in the calculus of variations , 1985 .

[10]  R. Rockafellar,et al.  Conjugate convex functions in optimal control and the calculus of variations , 1970 .

[11]  R. T. Rockafellar,et al.  The Euler and Weierstrass conditions for nonsmooth variational problems , 1996 .

[12]  R. Rockafellar.  Existence and duality theorems for convex problems of Bolza , 1971 .

[13]  B. Mordukhovich.  Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings , 1994 .

[14]  R. Rockafellar.  Equivalent Subgradient Versions of Hamiltonian and Euler--Llagrange Equations in Variational Analysis , 1996 .

[15]  F. Clarke.  Hamiltonian analysis of the generalized problem of Bolza , 1987 .

[16]  Philip D. Loewen,et al.  The adjoint arc in nonsmooth optimization , 1991 .

[17]  Philip D. Loewen.  Optimal Control Via Nonsmooth Analysis , 1993 .

[18]  B. Mordukhovich.  Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions , 1993 .