A characteristic method for fully convex Bolza problems over arcs of bounded variation. (English) Zbl 1420.49037
Summary: The aim of this paper is to study the value function of a fully convex Bolza problem with state constraints and under no coercivity assumptions. This requires that the state trajectories be of bounded variation rather than merely absolutely continuous. Our approach is based on the duality theory of classical convex analysis, and we establish a Fenchel-Young type equality between the value function of the Bolza problem and a suitable value function associated with its dual problem. The main result we present in this paper is a characteristic method that describes the evolution of the subgradients of the associated value functions. A few simple examples are provided to demonstrate the theory.
MSC:
| 49N15 | Duality theory (optimization) |
| 49N25 | Impulsive optimal control problems |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
Keywords:
fully convex optimal control; convex value functions; nonsmooth Hamiltonian systems; extended method of characteristics; impulsive systemsReferences:
| [1] | J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, New York, 1984. · Zbl 0538.34007 |
| [2] | J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York, 1984. · Zbl 0641.47066 |
| [3] | A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer-Verlag, New York, 2003. · Zbl 1017.49001 |
| [4] | F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer, New York, 2013. · Zbl 1277.49001 |
| [5] | F. Clarke, Y. Ledyaev, R. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998. · Zbl 1047.49500 |
| [6] | E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, London, 1955. · Zbl 0064.33002 |
| [7] | R. Goebel, Regularity of the optimal feedback and the value function in convex problems of optimal control, Set-Valued Anal., 12 (2004), pp. 127-145. · Zbl 1046.49020 |
| [8] | R. Goebel, Stationary Hamilton-Jacobi equations for convex control problems: Uniqueness and duality of solutions, in Optimal Control, Stabilization and Nonsmooth Analysis, Springer, New York, 2004, pp. 313-322. · Zbl 1104.49004 |
| [9] | R. Goebel, Convex optimal control problems with smooth Hamiltonians, SIAM J. Control Optim., 43 (2005), pp. 1787-1811, https://doi.org/10.1137/S0363012902411581. · Zbl 1072.49025 |
| [10] | R. Goebel, Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 357 (2005), pp. 2187-2203. · Zbl 1057.49026 |
| [11] | R. Goebel and R. T. Rockafellar, Generalized conjugacy in Hamilton-Jacobi theory for fully convex Lagrangians, J. Convex Anal., 9 (2002), pp. 463-473. · Zbl 1023.49023 |
| [12] | R. Goebel and R. T. Rockafellar, Linear-convex control and duality, in Geometric Control and Nonsmooth Analysis: In Honor of the 73rd Birthday of H. Hermes and of the 71st Birthday of R.T. Rockafellar, World Scientific, Hackensack, NJ, 2008, pp. 280-299. · Zbl 1206.49036 |
| [13] | R. Goebel and M. Subbotin, Continuous time linear quadratic regulator with control constraints via convex duality, IEEE Trans. Automat. Control, 52 (2007), pp. 886-892. · Zbl 1366.90161 |
| [14] | R. F. Hartl, S. P. Sethi, and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Rev., 37 (1995), pp. 181-218, https://doi.org/10.1137/1037043. · Zbl 0832.49013 |
| [15] | C. Hermosilla and P. R. Wolenski, Self-dual approximations to fully convex impulsive systems, in Proceedings of the 55th IEEE Conference on Decision and Control, IEEE, Washington, DC, 2016, pp. 6198-6203. |
| [16] | C. Hermosilla and P. R. Wolenski, Constrained and impulsive linear quadratic control problems, in Proceedings of the 20th IFAC World Congress, vol. 50, Toulouse, France, 2017, pp. 1637-1642. |
| [17] | T. Pennanen and A.-P. Perkkiö, Duality in convex problems of Bolza over functions of bounded variation, SIAM J. Control Optim., 52 (2014), pp. 1481-1498, https://doi.org/10.1137/130917612. · Zbl 1297.49060 |
| [18] | R. T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl., 32 (1970), pp. 174-222. · Zbl 0218.49004 |
| [19] | R. T. Rockafellar, Convex Analysis, Princeton Math. Ser. 28, Princeton University Press, Princeton, NJ, 1970. · Zbl 0193.18401 |
| [20] | R. T. Rockafellar, Generalized Hamiltonian equations for convex problems of Lagrange, Pacific J. Math., 33 (1970), pp. 411-427. · Zbl 0199.43002 |
| [21] | R. T. Rockafellar, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc., 159 (1971), pp. 1-40. · Zbl 0255.49007 |
| [22] | R. T. Rockafellar, Integrals which are convex functionals II, Pacific J. Math., 39 (1971), pp. 439-469. · Zbl 0236.46031 |
| [23] | R. T. Rockafellar, State constraints in convex control problems of Bolza, SIAM J. Control, 10 (1972), pp. 691-715, https://doi.org/10.1137/0310051. · Zbl 0224.49003 |
| [24] | R. T. Rockafellar, Saddle points of Hamiltonian systems in convex problems of Lagrange, J. Optim. Theory Appl., 12 (1973), pp. 367-390. · Zbl 0248.49016 |
| [25] | R. T. Rockafellar, Conjugate Duality and Optimization, CBMS-NSF Regional Conf. Ser. in Appl. Math. 16, SIAM, Philadelphia, 1974, https://doi.org/10.1137/1.9781611970524. · Zbl 0296.90036 |
| [26] | R. T. Rockafellar, Dual problems of Lagrange for arcs of bounded variation, in Calculus of Variations and Control Theory (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; Dedicated to Laurence Chisholm Young on the Occasion of his 70th Birthday), Academic Press, New York, 1976, pp. 155-192. · Zbl 0352.49005 |
| [27] | R. T. Rockafellar, Linear-quadratic programming and optimal control, SIAM J. Control Optim., 25 (1987), pp. 781-814, https://doi.org/10.1137/0325045. · Zbl 0617.49010 |
| [28] | R. T. Rockafellar, Hamiltonian trajectories and duality in the optimal control of linear systems with convex costs, SIAM J. Control Optim., 27 (1989), pp. 1007-1025, https://doi.org/10.1137/0327054. · Zbl 0682.49019 |
| [29] | R. T. Rockafellar, Hamilton-Jacobi theory and parametric analysis in fully convex problems of optimal control, J. Global Optim., 28 (2004), pp. 419-431. · Zbl 1083.49022 |
| [30] | R. T. Rockafellar and R. Wets, Variational Analysis, Springer-Verlag, New York, 2009. |
| [31] | R. T. Rockafellar and P. R. Wolenski, Convexity in Hamilton-Jacobi theory I: Dynamics and duality, SIAM J. Control Optim., 39 (2000), pp. 1323-1350, https://doi.org/10.1137/S0363012998345366. · Zbl 0998.49018 |
| [32] | R. T. Rockafellar and P. R. Wolenski, Convexity in Hamilton-Jacobi theory II: Envelope representations, SIAM J. Control Optim., 39 (2000), pp. 1351-1372, https://doi.org/10.1137/S0363012998345378. · Zbl 0998.49019 |
| [33] | R. Vinter, Optimal Control, Springer, New York, 2010. |
| [34] | C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, 2002. · Zbl 1023.46003 |
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