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Second-Order Sufficient Optimality Conditions for an Optimal Control Problem with Mixed Constraints

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Abstract

In this paper, we study second-order necessary and sufficient optimality conditions for an optimal control problem with state-control constraints. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we derive second-order necessary optimality conditions for an optimal control problem. In some cases, by using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain “no-gap” between second-order optimality conditions. Besides, second-order sufficient optimality conditions for an optimal control problem where objective function may only depend on the state, are also discussed.

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References

  1. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  2. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  3. Bonnans, J.F., Vega, C., Dupuis, X.: First and second-order optimality conditions for optimal control problems of state constrained integral equations. J. Optim. Theory Appl. 159, 1–40 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)

    Google Scholar 

  5. Casas, E., De los Reyes, J.C., Trältzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19, 616–643 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cernea, A., Frankowska, H.: A connection between the maximum principle and dynamic programming for constrained control problems. SIAM J. Optim. 44, 673–703 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cesari, L.: Optimization-Theory and Applications. Springer, New York (1983)

    Book  MATH  Google Scholar 

  8. Chieu, N.H., Kien, B.T., Toan, N.T.: Further results on subgradients of the value function to a parametric optimal control problem. J. Optim. Theory Appl. 168, 785–801 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dontchev, A.L.: Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  11. Frankowska, H., Hoehener, D., Tonon, D.: A second-order maximum principle in optimal control under state constraints. Serdica Math. J. 39, 233–270 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Frankowska, H., Tonon, D.: Pointwise second-order necessary optimality conditions for the Mayer problem with control contraints. SIAM J. Control Optim. 51, 3814–3843 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hilscher, R., Zeidan, V.: Second-order sufficiency criteria for a discrete optimal control problem. J. Abs. Differ. Equ. Appl. 8, 573–602 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hilscher, R., Zeidan, V.: Discrete optimal control: second-order optimality conditions. J. Abs. Differ. Equ. Appl. 8, 875–896 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hoehener, D.: Variational approach to second-order optimality conditions for control problems with pure state constraints. SIAM J. Control Optim. 50, 1139–1173 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, North-Holland (1979)

    Google Scholar 

  18. Kien, B.T., Nhu, V.H.: Second-order necessary optimality conditions for a class of semilinear elliptic optimal control problems with mixed pointwise constraints. SIAM J. Control Optim. 52, 1166–1202 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kien, B.T., Toan, N.T., Wong, M.M., Yao, J.-C.: Lower semicontinuity of the solution set to a parametric optimal control problem. SIAM J. Control Optim. 50, 2889–2906 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, L., Gao, Y., Wang, H.: Second-order sufficient optimality conditions for hybrid control problems with state jump. J. Ind. Manag. Optim. 11, 329–343 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Malanowski, K., Maurer, H., Pickenhain, S.: Second-order sufficient conditions for state-constrained optimal control problems. J. Optim. Theory Appl. 123, 595–617 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mangasarian, O.L., Shiau, T.H.: Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. SIAM J. Control Optim. 25, 583–595 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Marinkovíc, B.: Optimality conditions in discrete optimal control problem. J. Optim. Methsof. 22, 959–969 (2007)

    MATH  Google Scholar 

  24. Marinkovíc, B.: Optimality conditions for discrete optimal control problems with equality and inequality type constraints. Positivity 12, 535–545 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Marinkovíc, B.: Second-order optimality conditions in a discrete optimal control problem. Optimization 57, 539–548 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I, Basis Theory. Springer, Berlin (2006)

    Book  Google Scholar 

  27. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II, Applications. Springer, Berlin (2006)

    Google Scholar 

  28. Moussaoui, M., Seeger, A.: Epsilon-maximum principle of Pontryagin type and perturbation analysis of convex optimal control problems. SIAM J. Control Optim. 34, 407–427 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Páles, Z., Zeidan, V.: Optimum problems with certain lower semicontinuous set-valued constraints. SIAM J. Control Optim. 8, 707–727 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. Páles, Z., Zeidan, V.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32, 1476–1502 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Páles, Z., Zeidan, V.: Characterization of \(L^1\)-closed decomposable sets in \(L^{\infty }\). J. Math. Anal. Appl. 238, 491–515 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Páles, Z., Zeidan, V.: Optimal control problems with set-valued control and state constraints. SIAM J. Control Optim. 14, 334–358 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  34. Rockafellar, R.T., Wolenski, P.R.: Convexity in Hamilton-Jacobi theory, I: dynamics and duality. SIAM J. Control Optim. 39, 1323–1350 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rockafellar, R.T., Wolenski, P.R.: Convexity in Hamilton–Jacobi theory, II: envelope representation. SIAM J. Control Optim. 39, 1351–1372 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Thuy, L.Q., Thanh, B.T., Toan, N.T.: On the no-gap second-order optimality conditions for a discrete optimal control problem with mixed constraints. J. Optim. Theory Appl. 173, 421–442 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Toan, N.T., Kien, B.T.: Subgradients of the value function to a parametric optimal control problem. Set Valued Var. Anal. 18, 183–203 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Toan, N.T.: Mordukhovich subgradients of the value function to a parametric optimal control problem. Taiwan. J. Math. 19, 1051–1072 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Toan, N.T., Thuy, L.Q.: Second-order necessary optimality conditions for a discrete optimal control problem with mixed constraints. J. Glob. Optim. 64, 533–562 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Toan, N.T., Thuy, L.Q.: Second-order necessary optimality conditions for a discrete optimal control problem with nonlinear state equations. Optim. Control Appl. Meth. 41, 2250–2281 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  41. Toan, N.T., Thuy, L.Q.: Second-order necessary optimality conditions for an optimal control problem. Taiwan. J. Math. 24, 225–264 (2020)

    MathSciNet  MATH  Google Scholar 

  42. Toan, N.T., Thuy, L.Q.: Second-order necessary optimality conditions for an optimal control problem with nonlinear state equations. Positivity (2022). https://doi.org/10.1007/s11117-022-00898-x

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam

    L. Q. Thuy & N. T. Toan

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  1. L. Q. Thuy
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Thuy, L.Q., Toan, N.T. Second-Order Sufficient Optimality Conditions for an Optimal Control Problem with Mixed Constraints. Results Math 77, 208 (2022). https://doi.org/10.1007/s00025-022-01744-y

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