Feasible Perturbations of Control Systems with Pure State Constraints and Applications to Second-Order Optimality Conditions
Tóm tắt
We propose second-order necessary optimality conditions for optimal control problems with very general state and control constraints which hold true under weak regularity assumptions on the data. In particular the pure state constraints are general closed sets, the optimal control is supposed to be merely measurable and the dynamics may be discontinuous in the time variable as well. These results are obtained by an approach based on local perturbations of the reference process by second-order tangent directions. This method allows direct and quite simple proofs.
Tài liệu tham khảo
Athans, M., Falb, P.L.: Optimal Control, an Introduction to the Theory and Its Applications. McGraw-Hill, New York (1966)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis, Systems Control Found. Appl. Birkhäuser, Boston (1990)
Auslender, A.: Stability in mathematical programming with nondifferentiable data. SIAM J. Control Optim. 22, 239–254 (1984)
Ben-Tal, A., Zowe, J.: A unified theory of first and second order conditions for extremum problems in topological vector spaces. In: Optimality and Stability in Mathematical Programming. Springer, Berlin (1982)
Bettiol, P., Frankowska, H.: Normality of the maximum principle for nonconvex constrained Bolza problems. J. Differ. Equ. 243, 256–269 (2007)
Bonnans, J.F., Hermant, A.: No-gap second-order optimality conditions for optimal control problems with a single state constraint and control. Math. Program. 117, 21–50 (2009)
Bonnans, J.F., Hermant, A.: Revisiting the analysis of optimal control problems with several state constraints. Control Cybern. 38, 1021–1052 (2009)
Bonnans, J.F., Hermant, A.: Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, 561–598 (2009)
Borisovich, Y.G., Gel’man, B.D., Myshkis, A.D., Obukhovskii, V.V.: Multivalued mappings. J. Sov. Math. 24, 719–791 (1984)
Bryson, A.E., Ho, Y.-C.: Applied Optimal Control. Ginn, Needham Heights (1969)
Cernea, A., Frankowska, H.: A connection between the maximum principle and dynamic programming for constrained optimal control problems. SIAM J. Control Optim. 44, 673–703 (2005)
Chaney, R.W.: Second-order directional derivatives for nonsmooth functions. J. Math. Anal. Appl. 128, 495–511 (1987)
Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)
Clarke, F.H., de Pinho, M.R.: The nonsmooth maximum principle. Control Cybern. 38, 1151–1167 (2009)
Cominetti, R.: Metric regularity, tangent sets and second order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)
Cominetti, R., Correa, R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28, 789–809 (1990)
de Pinho, M.R., Rosenblueth, J.F.: Mixed constraints in optimal control: an implicit function approach. IMA J. Math. Control Inf. 24, 197–218 (2007)
Dmitruk, A.V.: Jacobi type conditions for singular extremals. Control Cybern. 37, 285–306 (2008)
Dubovitskii, A.Y., Milyutin, A.A.: Extremum problems with constraints. Dokl. Akad. Nauk SSSR 149, 759–762 (1963)
Frankowska, H.: Regularity of minimizers and of adjoint states in optimal control under state constraints. J. Convex Anal. 13, 299–328 (2006)
Frankowska, H.: Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints. Control Cybern. 38, 1327–1340 (2009)
Frankowska, H., Mazzola, M.: On relations of the adjoint state to the value function for optimal control problems with state constraints. Nonlinear Differ. Equ. Appl. (2012). doi:10.1007/s00030-012-0183-0
Galbraith, G.N., Vinter, R.B.: Lipschitz continuity of optimal controls for state constrained problems. SIAM J. Control Optim. 42, 1727–1744 (2003)
Gilbert, E.G., Bernstein, D.S.: Second-order necessary conditions in optimal control: accessory-problem results without normality conditions. J. Optim. Theory Appl. 41, 75–106 (1983)
Hiriart-Urruty, J.-B., Strodiot, J.-J., Nguyen, V.H.: Generalized Hessian matrix and second-order optimality conditions for problems with C 1,1 data. Appl. Math. Optim. 11, 43–56 (1984)
Hoehener, D.: Second-order optimality conditions for a Bolza problem with mixed constraints. In: Bittanti, S., Cenedese, A., Zampieri, S. (eds.) Proceedings of the 18th IFAC World Congress, pp. 2594–2599 (2011)
Hoehener, D.: Variational approach to second-order optimality conditions for control problems with pure state constraints. SIAM J. Control Optim. 50, 1139–1173 (2012)
Kawasaki, H.: An envelope-like effect on infinitely many inequality constraints on second-order necessary conditions for minimization problems. Math. Program. 41, 73–96 (1988)
Kawasaki, H.: Second order necessary optimality conditions for minimizing a sup-type function. Math. Program. 49, 213–229 (1991)
Maurer, H., Zowe, J.: First and second-order necessary and sufficient optimality conditions for infinite dimensional programming problems. Math. Program. 16, 98–110 (1979)
Milyutin, A.A., Osmolovskii, N.P.: Calculus of Variations and Optimal Control. American Mathematical Society, Providence (1998)
Osmolovskii, N.P.: Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. J. Math. Sci. (N.Y.) 123, 3987–4122 (2004)
Osmolovskii, N.P.: Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J. Math. Sci. (N.Y.) 173, 1–106 (2011)
Páles, Z., Zeidan, V.: First- and second-order necessary conditions for control problems with constraints. Trans. Am. Math. Soc. 346, 421–453 (1994)
Páles, Z., Zeidan, V.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32, 1476–1502 (1994)
Páles, Z., Zeidan, V.: Generalized Hessian for C 1,1 functions in infinite dimensional normed spaces. Math. Program. 74, 59–78 (1996)
Páles, Z., Zeidan, V.: Optimum problems with certain lower semicontinuous set-valued constraints. SIAM J. Optim. 8, 707–727 (1998)
Páles, Z., Zeidan, V.: Optimal control problems with set-valued control and state constraints. SIAM J. Optim. 14, 334–358 (2003)
Páles, Z., Zeidan, V.: First- and second-order optimality conditions for a strong local minimum in control problems with pure state constraints. Nonlinear Anal. 67, 2506–2526 (2007)
Rampazzo, F., Vinter, R.B.: A theorem on existence of neighboring trajectories satisfying a state constraint, with applications to optimal control. IMA J. Math. Control Inf. 16, 335–351 (1999)
Ritter, K.: Optimization theory in linear spaces I. Math. Ann. 182, 189–206 (1969)
Robinson, S.M.: Stability theorems for systems of inequalities, part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)
Rockafellar, R.T.: Integrals which are convex functionals II. Pac. J. Math. 39, 439–469 (1971)
Rockafellar, R.T.: First- and second-order epi-differentiability in nonlinear programming. Trans. Am. Math. Soc. 307, 75–108 (1988)
Stefani, G., Zezza, P.: Optimality conditions for a constrained control problem. SIAM J. Control Optim. 34, 635–659 (1996)
Vinter, R.B.: Optimal Control. Birkhäuser, Boston (2000)
Warga, J.: A second-order Lagrangian condition for restricted control problems. J. Optim. Theory Appl. 24, 475–483 (1978)
Warga, J.: A second-order condition that strengthens Pontryagin’s maximum principle. J. Differ. Equ. 28, 284–307 (1979)