# American Institute of Mathematical Sciences

March  2013, 18(2): 331-348. doi: 10.3934/dcdsb.2013.18.331

## Optimal control of ODE systems involving a rate independent variational inequality

 1 Fakultät für Mathematik, TU München, Boltzmannstr. 3, D 85747 Garching bei München, Germany 2 Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1

Received  December 2011 Revised  April 2012 Published  November 2012

This paper is concerned with an optimal control problem for a system of ordinary differential equations with rate independent hysteresis modelled as a rate independent evolution variational inequality with a closed convex constraint $Z\subset \mathbb{R}^m$. We prove existence of optimal solutions as well as necessary optimality conditions of first order. In particular, under certain regularity assumptions we completely characterize the jump behaviour of the adjoint.
Citation: Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331
##### References:
 [1] J.-J. Moreau, Problème d'evolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794. [2] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq., 26 (1977), 347-374. [3] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in "Nonlinear Differential Equations" (eds. P. Drábek, P. Krejčí and P. Takáč),Research Notes in Mathematics 404, Chapman & Hall CRC, London, (1999), 47-110. [4] A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994. [5] P. Krejčí and Ph. Laurençcot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. [6] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 89-176. [7] M. Brokate, "Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ," Verlag Peter D. Lang, Frankfurt am Main, 1987. [8] M. Brokate, Optimal control of ODE systems with hysteresis nonlinearities, in "Trends in Mathematical Optimization (Irsee, 1986)" Internat. Schriftenreihe Numer. Math. 84, Birkhäuser, Basel, (1988), 25-41. [9] M. Brokate, ODE control problems including the Preisach hysteresis operator: Necessary optimality conditions, in "Dynamic Economic Models and Optimal Control (Vienna, 1991)" North-Holland, Amsterdam, (1992), 51-68. [10] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'' Nauka, Moscow, 1983. (In Russian.) [11] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'' Springer, Heidelberg, 1989. [12] M. Brokate, Optimal control of systems described by ordinary differential equations with nonlinear characteristics of hysteresis type I., Translated from the German and with a Preface by V. B. Kolmanovskiĭ and N. I. Koroleva, Avtomat. i Telemekh., (1991), 89-176; Automat. Remote Control., 52 (1991), 1639-1681. [13] M. Brokate, Optimal control of systems described by ordinary differential equations with nonlinear characteristics of hysteresis type II., Avtomat. i Telemekh., (1992), 2-40; Automat. Remote Control., 53 (1992), 1-33. [14] A. Bensoussan, K. Chandrasekharan and J. Turi, Optimal control of variational inequalities, Commun. Inf. Syst., 10 (2010), 203-220. [15] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 117-159. [16] F. Bagagiolo, An infinite horizon optimal control problem for some switching systems, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 443-462. [17] A. Gudovich and M. Quincampoix, Optimal control with hysteresis nonlinearity and multidimensional play operator, SIAM J. Control Opt., 49 (2011), 788-807. doi: 10.1137/090770011. [18] F. Bagagiolo and M. Benetton, About an optimal visiting problem, Appl. Math. Optim., 65 (2012), 31-51. [19] R. B. Holmes, Smoothness of certain metric projections on Hilbert space, Trans. Amer. Math. Soc., 184 (1973), 87-100. [20] S. Fitzpatrick and R. R. Phelps, Differentiability of the metric projection in Hilbert space, Trans. Amer. Math. Soc., 270 (1982), 483-501. [21] M. C. Delfour and J.-P. Zolesio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization,'' SIAM, Philadelphia, 2001.

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##### References:
 [1] J.-J. Moreau, Problème d'evolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794. [2] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq., 26 (1977), 347-374. [3] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in "Nonlinear Differential Equations" (eds. P. Drábek, P. Krejčí and P. Takáč),Research Notes in Mathematics 404, Chapman & Hall CRC, London, (1999), 47-110. [4] A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994. [5] P. Krejčí and Ph. Laurençcot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. [6] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 89-176. [7] M. Brokate, "Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ," Verlag Peter D. Lang, Frankfurt am Main, 1987. [8] M. Brokate, Optimal control of ODE systems with hysteresis nonlinearities, in "Trends in Mathematical Optimization (Irsee, 1986)" Internat. Schriftenreihe Numer. Math. 84, Birkhäuser, Basel, (1988), 25-41. [9] M. Brokate, ODE control problems including the Preisach hysteresis operator: Necessary optimality conditions, in "Dynamic Economic Models and Optimal Control (Vienna, 1991)" North-Holland, Amsterdam, (1992), 51-68. [10] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'' Nauka, Moscow, 1983. (In Russian.) [11] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'' Springer, Heidelberg, 1989. [12] M. Brokate, Optimal control of systems described by ordinary differential equations with nonlinear characteristics of hysteresis type I., Translated from the German and with a Preface by V. B. Kolmanovskiĭ and N. I. Koroleva, Avtomat. i Telemekh., (1991), 89-176; Automat. Remote Control., 52 (1991), 1639-1681. [13] M. Brokate, Optimal control of systems described by ordinary differential equations with nonlinear characteristics of hysteresis type II., Avtomat. i Telemekh., (1992), 2-40; Automat. Remote Control., 53 (1992), 1-33. [14] A. Bensoussan, K. Chandrasekharan and J. Turi, Optimal control of variational inequalities, Commun. Inf. Syst., 10 (2010), 203-220. [15] G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 117-159. [16] F. Bagagiolo, An infinite horizon optimal control problem for some switching systems, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 443-462. [17] A. Gudovich and M. Quincampoix, Optimal control with hysteresis nonlinearity and multidimensional play operator, SIAM J. Control Opt., 49 (2011), 788-807. doi: 10.1137/090770011. [18] F. Bagagiolo and M. Benetton, About an optimal visiting problem, Appl. Math. Optim., 65 (2012), 31-51. [19] R. B. Holmes, Smoothness of certain metric projections on Hilbert space, Trans. Amer. Math. Soc., 184 (1973), 87-100. [20] S. Fitzpatrick and R. R. Phelps, Differentiability of the metric projection in Hilbert space, Trans. Amer. Math. Soc., 270 (1982), 483-501. [21] M. C. Delfour and J.-P. Zolesio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization,'' SIAM, Philadelphia, 2001.
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