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 & 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). [2] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Diff. Eq., 26 (1977), 347. [3] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators,, in, 404 (1999), 47. [4] A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). [5] P. Krejčí and Ph. Laurençcot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159. [6] P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 89. [7] M. Brokate, "Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ,", Verlag Peter D. Lang, (1987). [8] M. Brokate, Optimal control of ODE systems with hysteresis nonlinearities,, in, 84 (1988), 25. [9] M. Brokate, ODE control problems including the Preisach hysteresis operator: Necessary optimality conditions,, in, (1992), 51. [10] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'', Nauka, (1983). [11] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'', Springer, (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, 52 (1991), 89. [13] M. Brokate, Optimal control of systems described by ordinary differential equations with nonlinear characteristics of hysteresis type II.,, Avtomat. i Telemekh., 53 (1992), 2. [14] A. Bensoussan, K. Chandrasekharan and J. Turi, Optimal control of variational inequalities,, Commun. Inf. Syst., 10 (2010), 203. [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. [16] F. Bagagiolo, An infinite horizon optimal control problem for some switching systems,, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 443. [17] A. Gudovich and M. Quincampoix, Optimal control with hysteresis nonlinearity and multidimensional play operator,, SIAM J. Control Opt., 49 (2011), 788. doi: 10.1137/090770011. [18] F. Bagagiolo and M. Benetton, About an optimal visiting problem,, Appl. Math. Optim., 65 (2012), 31. [19] R. B. Holmes, Smoothness of certain metric projections on Hilbert space,, Trans. Amer. Math. Soc., 184 (1973), 87. [20] S. Fitzpatrick and R. R. Phelps, Differentiability of the metric projection in Hilbert space,, Trans. Amer. Math. Soc., 270 (1982), 483. [21] M. C. Delfour and J.-P. Zolesio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization,'', SIAM, (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). [2] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Diff. Eq., 26 (1977), 347. [3] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators,, in, 404 (1999), 47. [4] A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). [5] P. Krejčí and Ph. Laurençcot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159. [6] P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 89. [7] M. Brokate, "Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ,", Verlag Peter D. Lang, (1987). [8] M. Brokate, Optimal control of ODE systems with hysteresis nonlinearities,, in, 84 (1988), 25. [9] M. Brokate, ODE control problems including the Preisach hysteresis operator: Necessary optimality conditions,, in, (1992), 51. [10] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'', Nauka, (1983). [11] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,'', Springer, (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, 52 (1991), 89. [13] M. Brokate, Optimal control of systems described by ordinary differential equations with nonlinear characteristics of hysteresis type II.,, Avtomat. i Telemekh., 53 (1992), 2. [14] A. Bensoussan, K. Chandrasekharan and J. Turi, Optimal control of variational inequalities,, Commun. Inf. Syst., 10 (2010), 203. [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. [16] F. Bagagiolo, An infinite horizon optimal control problem for some switching systems,, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 443. [17] A. Gudovich and M. Quincampoix, Optimal control with hysteresis nonlinearity and multidimensional play operator,, SIAM J. Control Opt., 49 (2011), 788. doi: 10.1137/090770011. [18] F. Bagagiolo and M. Benetton, About an optimal visiting problem,, Appl. Math. Optim., 65 (2012), 31. [19] R. B. Holmes, Smoothness of certain metric projections on Hilbert space,, Trans. Amer. Math. Soc., 184 (1973), 87. [20] S. Fitzpatrick and R. R. Phelps, Differentiability of the metric projection in Hilbert space,, Trans. Amer. Math. Soc., 270 (1982), 483. [21] M. C. Delfour and J.-P. Zolesio, "Shapes and Geometries. Analysis, Differential Calculus and Optimization,'', SIAM, (2001).
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