Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1705-1734.

The main focus of this paper is to develop a sufficiency criterion for optimality in nonlinear optimal control problems defined on time scales. In particular, it is shown that the coercivity of the second variation together with the controllability of the linearized dynamic system are sufficient for the weak local minimality. The method employed is based on a direct approach using the structure of this optimal control problem. The second aim pertains to the sensitivity analysis for parametric control problems defined on time scales with separately varying state endpoints. Assuming a slight strengthening of the sufficiency criterion at a base value of the parameter, the perturbed problem is shown to have a weak local minimum and the corresponding multipliers are shown to be continuously differentiable with respect to the parameter. A link is established between (i) a modification of the shooting method for solving the associated boundary value problem, and (ii) the sufficient conditions involving the coercivity of the accessory problem, as opposed to the Riccati equation, which is also used for this task. This link is new even for the continuous time setting.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017070
Classification : 49K15, 49K40, 34N05, 34K35, 90C31, 39A12
Mots clés : Optimal control problem on time scales, Weak Pontryagin maximum principle, Weak local minimum, Coercivity, Sufficient optimality condition, Sensitivity analysis, Second variation, Controllability
Šimon Hilscher, Roman 1 ; Zeidan, Vera 1

1
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     title = {Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity},
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Šimon Hilscher, Roman; Zeidan, Vera. Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1705-1734. doi : 10.1051/cocv/2017070. http://www.numdam.org/articles/10.1051/cocv/2017070/

[1] R.P. Agarwal, V. Otero–Espinar and K. Perera, Basic properties of Sobolev’s spaces on time scales. Adv. Differ. Equ. 2006 (2006) 38121. | DOI | MR | Zbl

[2] M.S. Aronna, J.F. Bonnans, A.V. Dmitruk and P.A. Lotito, Quadratic order conditions for bang-singular extremals. Numer. Algebra Control Optim. 2 (2012) 511–546. | DOI | MR | Zbl

[3] F.M. Atici, D.C. Biles and A. Lebedinsky, A utility maximisation problem on multiple time scales. Int. J. Dyn. Syst. Differ. Equ. 3 (2011) 38–47. | MR | Zbl

[4] B. Aulbach and L. Neidhart, Integration on measure chains. In: “New Progress in Difference Equations”, Proc. of the Sixth International Conference on Difference Equations (Augsburg, 2001). Edited by B. Aulbach, S. Elaydi, and G. Ladas. Chapman & Hall/CRC, Boca Raton, FL (2004) 239–252. | DOI | MR | Zbl

[5] Z. Bartosiewicz and D.F.M. Torres, Noether’s theorem on time scales. J. Math. Anal. Appl. 342 (2008) 1220–1226. | DOI | MR | Zbl

[6] M. Bohner, Calculus of variations on time scales. Dynam. Sys. Appl. 13 (2004) 339–349. | MR | Zbl

[7] M. Bohner, M. Fan and J. Zhang, Existence of periodic solutions in predator-prey and competition dynamic systems. Nonlinear Anal. Real World Appl. 7 (2006) 1193–1204. | DOI | MR | Zbl

[8] M. Bohner, M. Fan and J. Zhang, Periodicity of scalar dynamic equations and applications to population models. J. Math. Anal. Appl. 330 (2007) 1–9. | DOI | MR | Zbl

[9] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Boston 2001. | MR | Zbl

[10] M. Bohner, A. Peterson and editors, Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston 2003. | DOI | Zbl

[11] M. Bohner and N. Wintz, The linear quadratic regulator on time scales. Int. J. Differ. Equ. 5 (2010) 149–174. | MR

[12] M. Bohner and N. Wintz, The linear quadratic tracker on time scales. Int. J. Dyn. Syst. Differ. Equ. 3 (2011) 423–447. | MR | Zbl

[13] M. Bohner and N. Wintz, The Kalman filter for linear systems on time scales. J. Math. Anal. Appl. 46 (2013) 419–436. | DOI | MR | Zbl

[14] V.G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming method. J. SIAM J. Control Optim. 4 (1966) 326–361. | DOI | MR | Zbl

[15] J.F. Bonnans, X. Dupuis and L. Pfeiffer, Second-order necessary conditions in Pontryagin form for optimal control problems. SIAM J. Control Optim. 52 (2014) 3887–3916. | DOI | MR | Zbl

[16] L. Bourdin and E. Trélat, Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales. SIAM J. Control Optim. 51 (2013) 3781–3813. | DOI | MR | Zbl

[17] L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales. Math. Control. Relat. Fields 6 (2016) 53–94. | DOI | MR | Zbl

[18] J.V. Breakwell and Y.C. Ho, On the conjugate point condition for the control problem. Internat. J. Engrg. Sci. 2 (1964/1965) 565–579. | DOI | MR | Zbl

[19] J.V. Breakwell, J.L. Speyer and A.E. Bryson, Optimization and control of nonlinear systems using the second variation. SIAM J. Control Optim. 1 (1963) 193–223. | Zbl

[20] P.P. Cai,J.L. Fu and Y.X. Guo, Noether symmetries of the nonconservative and nonholonomic systems on time scales. Science China – Physics Mechanics & Astronomy 56 (2013) 1017–1028. | DOI

[21] A.L. Donchev and W.W. Hager, Lipschitzian stability in nonlinear control and optimization. SIAM J. Control Optim. 31 (1993) 569–603. | DOI | MR | Zbl

[22] A.L. Donchev, W.W. Hager, A.B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim. 31 (1995) 297–326. | DOI | MR | Zbl

[23] A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Vol. 165 of Mathematics in Science and Engineering. Academic Press, Orlando, FL (1983) | MR | Zbl

[24] G.S. Guseinov, Integration on time scales. J. Math. Anal. Appl. 285 (2003) 107–127. | DOI | MR | Zbl

[25] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus. Results Math. 18 (1990) 18–56. | DOI | MR | Zbl

[26] R. Hilscher, W. Kratz and V. Zeidan, Differentiation of solutions of dynamic equations on time scales with respect to parameters. Adv. Dyn. Syst. Appl. 4 (2009) 35–54. | MR

[27] R. Hilscher and V. Zeidan, Second order sufficiency criteria for a discrete optimal control problem. J. Differ. Equ. Appl. 8 (2002) 573–602. | DOI | MR | Zbl

[28] R. Hilscher and V. Zeidan, Calculus of variations on time scales: weak local piecewise C1rd solutions with variable endpoints. J. Math. Anal. Appl. 289 (2004) 143–166. | DOI | MR | Zbl

[29] R. Hilscher and V. Zeidan, Legendre, Jacobi, and Riccati type conditions for time scale variational problem with application. Dynam. Systems Appl. 16 (2007) 451–480. | MR | Zbl

[30] R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals. Analysis (Munich) 28 (2008) 1–28. | MR | Zbl

[31] R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal. 70 (2009) 3209–3226. | DOI | MR | Zbl

[32] V. Kac andP. Cheung, Quantum Calculus. Springer-Verlag, New York, 2002. | MR | Zbl

[33] R.E. Kalman, The theory of optimal control and the calculus of variations. In Symposium on Mathematical Optimization Techniques (Santa Monica, CA, 1960). Univ. California Press, Berkeley, CA. Mathematical Optimization Techniques. Edited by R. Bellman (1963) 309–331 | DOI | MR | Zbl

[34] J.-L. Lions, Optimal control of systems governed by partial differential equations. Translated from the French by S.K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170. Springer-Verlag, New York-Berlin (1971) | MR | Zbl

[35] D.G. Luenberger, Linear and Nonlinear Programming. 2nd ed. Addison-Wesley, Reading, MA, 1984. | Zbl

[36] K. Malanowski, Sensitivity analysis of optimization problems in Hilbert space with application to optimal control. Appl. Math. Optim. 21 (1990) 1–20. | DOI | MR | Zbl

[37] K. Malanowski and H. Maurer, Sensitivity analysis for parametric control problems with control-state constraints. Comput. Optim. Appl. 5 (1996) 253–283. | DOI | MR | Zbl

[38] A.B. Malinowska and D.F.M. Torres, Necessary and sufficient conditions for local Pareto optimality on time scales. J. Math. Sci. (N.Y.) 161 (2009) 803–810. | DOI | MR | Zbl

[39] A.B. Malinowska and D.F.M. Torres, Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales. Appl. Math. Comput. 217 (2010) 1158–1162. | MR | Zbl

[40] A.B. Malinowska and D.F.M. Torres, Euler–Lagrange equations for composition functionals in calculus of variations on time scales Discrete. Contin. Dyn. Syst. Ser. A 29 (2011) 577–593. | DOI | MR | Zbl

[41] A.B. Malinowska and D.F.M. Torres, Quantum Variational Calculus. Springer Briefs in Electrical and Computer Engineering. Springer, Cham (2014). | MR | Zbl

[42] H. Maurer and H.J. Pesch, Solution differentiability for nonlinear parametric control problem. SIAM J. Control Optim. 32 (1994) 1542–1554. | DOI | MR | Zbl

[43] H. Maurer and H.J. Pesch, Solution differentiability for parametric nonlinear control problems with control-state constraints. J. Optim. Theory Appl. 86, (1995) 285–309. | DOI | MR | Zbl

[44] A.A. Milyutin andN.P. Osmolovskii, Calculus of variations and optimal control. Translated from the Russian manuscript by Dimitrii Chibisov. Translations of Vol. 180 of Mathematical Monographs. American Mathematical Society, Providence, RI (1998). | MR | Zbl

[45] D. Orrell and V. Zeidan, Another Jacobi sufficiency criterion for optimal control with smooth constraints. J. Optim. Theory Appl. 58 (1988) 283–300. | DOI | MR | Zbl

[46] N.P. Osmolovskii Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J. Math. Sci. 173 (2011) 1–106. | DOI | MR | Zbl

[47] L. Poggiolini and G. Stefani, Bang-singular-bang extremals: sufficient optimality conditions. J. Dyn. Control Syst. 17 (2011) 469–514. | DOI | MR | Zbl

[48] B.P. Rynne L2 spaces and boundary value problems on time-scales. J. Math. Anal. Appl. 328 (2007) 1217–1236. | DOI | MR | Zbl

[49] J.L. Speyer and D.H. Jacobson, Primer on Optimal Control Theory. In Vol. 20 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2010). | MR | Zbl

[50] P. Šepitka and R. Šimon Hilscher, Principal solutions at infinity for time scale symplectic systems without controllability condition. J. Math. Anal. Appl. 444 (2016) 852–880. | DOI | MR | Zbl

[51] R. Šimon Hilscher A note on the time scale calculus of variations problems. In Vol. 14 of Ulmer Seminare über Funktionalanalysis und Differentialgleichungen. University of Ulm, Ulm (2009) 223–230

[52] R. Šimon Hilscher and V. Zeidan, Symplectic structure of Jacobi systems on time scales. Int. J. Differ. Equ. 5 (2010) 55–81. | MR

[53] R. Šimon Hilscher and V. Zeidan, First order conditions for generalized variational problems over time scales. Comput. Math. Appl. 62 (2011) 3490–3503. | DOI | MR | Zbl

[54] R. Šimon Hilscher and V. Zeidan Hamilton–Jacobi theory over time scales and applications to linear-quadratic problems. Nonlinear Anal. 75 (2012) 932–950. | DOI | MR | Zbl

[55] V. Zeidan, Continuous versus discrete nonlinear optimal control problems. In: Proc. of the 14th International Conference on Difference Equations and Applications (Istanbul, 2008). Edited by M. Bohner, Z. Došlá, G. Ladas, M. Ünal, and A. Zafer. Uğur-Bahçeşehir University Publishing Company, Istanbul (2009) 73–93. | MR | Zbl

[56] V. Zeidan, Constrained linear-quadratic control problems on time scales and weak normality. In Vol. 26 of Dynamic Systems and Applications (2017) 627–662 | Zbl

[57] V. Zeidan and P. Zezza, The conjugate point condition for smooth control sets. J. Math. Anal. Appl. 132 (1988) 572–589. | DOI | MR | Zbl

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