Abstract
First and second order necessary conditions of optimality for an impulsive control problem are presented and derived. One of the main features of these results is that no a priori normality assumptions are required and they are informative for abnormal control processes as well. This feature follows from the fact that the conditions are derived from an extremal principle, which is proved for an abstract minimization problem with equality and inequality type constraints and constraints given by convex cone.
Two simple examples illustrate the power of our result.
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REFERENCES
A. A. Agrachev and R. V. Gamkrelidze, Index of extremality and quasiextremality. Russian Math. Surveys 284 (1985), 11-14.
A. V. Arutyunov, Optimality conditions: Abnormal and degenerate problems. Kluwer Acad. Publ., 2000.
A. V. Arutyunov, Second-order conditions in extremal problems. The abnormal points. In: Trans. Amer. Math. Soc. 350 (1998), 4341-4365.
J.-P. Aubin and I. Ekeland, Applied nonlinear analysis. John Wiley and Sons, 1984.
A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields. J. Optim. Theory Appl. 71 (1991), No. 1 67-83.
A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions. J. Optim. Theory Appl. 81 (1994), No. 3, 435-457.
B. Brogliato Nonsmooth impact mechanics: Models, dynamics and control. Lect. Notes Control and Inform. Sci. 220, Springer-Verlag, 1996.
C. Clark, F. Clarke, and G. Munro, The optimal exploitation of renewable stocks. Econometrica 47 (1979), 25-47.
F. H. Clarke, Yu. Ledyaev, R. Stern, and P. Wolenski, Nonsmooth analysis and control theory. Grad. Texts in Math. 178, Springer-Verlag, New York, 1998.
G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls. Differential Integral Equations 4 (1991), 739-765.
V. Dykhta and O. Sumsonuk, A maximum principle for optimal impulsive processes and its applications. In: Proc. European Control Confer. V.2-FR-A-D-3, Brussels, Belgium, 1997.
V. Dykhta, Second order optimality conditions for impulse control problem and multiprocesses. In: Proc. of IFAC Workshop “Singular Solutions and Perturbations in Control Systems”, Pereslavl-Zalessky, Russia, July 7–11, 1997, Elsevier Science Ltd., UK, 1997, pp. 97-101.
G. Kolokolnikova, A variational maximum principle for discontinuous trajectories of unbounded asymptotically linear control systems. Diff. Uravn. 33, No. 12, 1631-1638. English transl. J. Differ. Equations 33 (1997), No. 12, 1633-1640.
D. F. Lawden, Optimal trajectories for space navigation. Butterworth, London, 1663.
U. Ledzewicz and H. Schaettler, Higher order conditions for optimality. SIAM J. Control Optim. 37 (1998), No. 1, 33-53.
U. Ledzewicz and H. Schaettler, Higher order extended maximum principles for optimal control problems with nonregular constraints. In: Optimal Control: Theory, Algorithms and Applications. (W. Hager and P. Pardalos, Eds.), Kluwer Academic Publ. (1998), pp. 298-325.
J. P. Marec, Optimal space trajectories. Elsevier, 1979.
B. M. Miller, Optimality conditions in problems of generalized control. Automat Remote Control 5 (1992), 50-58.
B. S. Mordukhovich, On necessary conditions for an extremum in nonsmooth optimization. Sov. Math. Dokl. 32 (1985), 215-220.
Yu. Orlov, Theory of optimal systems with generalized controls [Russian]. Moscow, Nauka, 1988.
F. Pereira, A maximum principle for impulsive control problems with state constraints. Internat. J. Comput. Appl. Math. 19 (2000), No. 2, 1-19.
F. Pereira and G. Silva, Necessary conditions of optimality for vector-valued impulsive control problems. In: Systems and Control Letters 40 (2000), 205-215.
R. T. Rockafellar, Optimality conditions for convex control problems with nonnegative states and the possibility of jumps. In: Game Theory and Math. Economics (O. Moeschlin, D. Pallaschke, Eds.). NorthHolland Publishing Company, 1981.
A. V. Sarychev, Optimization of generalized controls in a nonlinear time optimal problem. Differential Equations 27 (1991), No. 5, 539-550.
G. N. Silva and R. B. Vinter, Measure differential inclusions. J. Math. Anal. Appl. 202 (1996), 727-746.
G. N. Silva and R. B. Vinter, Necessary conditions for optimal impulsive control problems. SIAM J. Control Optim. 35 (1997), 1829-1846.
R. B. Vinter and F. M. L. Pereira, A maximum principle for optimal processes with discontinuous trajectories. SIAM J. Control Optim. 26 (1988), 205-229.
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Arutyunov, A., Jaćimović, V. & Pereira, F. Second Order Necessary Conditions for Optimal Impulsive Control Problems. Journal of Dynamical and Control Systems 9, 131–153 (2003). https://doi.org/10.1023/A:1022111402527
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DOI: https://doi.org/10.1023/A:1022111402527