Averaged control.(English)Zbl 1309.93029

Summary: We analyze the problem of controlling parameter-dependent systems. We introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter.
First, we consider the problem of controllability for linear finite-dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition, in the spirit of the classical rank condition for linear control systems, but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action.
We also describe some open problems and directions of possible research, in particular on the average controllability of evolution partial differential equations. In this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system.

MSC:

 93B05 Controllability 93B07 Observability 93C15 Control/observation systems governed by ordinary differential equations 93C20 Control/observation systems governed by partial differential equations 93C05 Linear systems in control theory
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References:

 [1] Ackermann, J., Robust control: the parameter space approach, (2002), Springer-Verlag London [2] Ammar-Khodja, F.; Benabdallah, A.; de Teresa, L.; González-Burgos, M., Recent results on the controllability of linear coupled parabolic problems: a survey, Mathematical Control and Related Fields, 3, 267-306, (2011) · Zbl 1235.93041 [3] Beck, J. V.; Blackwell, B.; Claire, C. R.St., Inverse heat conduction. ill-posed problems, (1985), J. Wiley & Sons New York · Zbl 0633.73120 [4] Brezis, H., (Functional analysis, sobolev spaces and partial differential equations, Universitext, (2011), Springer New York) · Zbl 1220.46002 [5] Casas, E.; Zuazua, E., Spike controls for elliptic and parabolic PDE, Systems & Control Letters, 62, 311-318, (2013) · Zbl 1267.49005 [6] Coron, J.-M., (Control and nonlinearity, Mathematical surveys and monographs, Vol. 136, (2007), American Mathematical Society Providence, RI) [7] Curtain, R.; Zwart, H., (An introduction to infinite-dimensional linear systems theory, Texts in applied mathematics, Vol. 21, (1995), Springer-Verlag New York) · Zbl 0839.93001 [8] Dáger, R.; Zuazua, E., (Wave propagation and control in $$1 - d$$ vibrating multi-structures, Mathématiques et applications, Vol. 50, (2006), Springer Verlag Berlin) · Zbl 1083.74002 [9] Dehman, B.; Léautaud, M.; Le Rousseau, J., Controllability of two coupled wave equations on a manifold, Archive for Rational Mechanics and Analysis, 211, 1, 113-187, (2014) · Zbl 1290.35278 [10] Ervedoza, S.; Zuazua, E., (On the numerical approximation of exact controls for waves, Springer briefs in mathematics, (2013), Springer New York) · Zbl 1282.93001 [11] Fabre, C.; Puel, J.; Zuazua, E., On the density of the range of the semigroup for semilinear heat equations, (Control and optimal design of distributed parameter systems, IMA vol. math. appl., Vol. 70, (1995), Springer New York), 73-92 · Zbl 0822.35075 [12] Fernández-Cara, E.; Zuazua, E., The cost of approximate controllability for heat equations: the linear case, Advances in Differential Equations, 5, 4-6, 465-514, (2000) · Zbl 1007.93034 [13] Fursikov, A. V.; Imanuvilov, O. Yu., (Controllability of evolution equations, Lecture notes series, Vol. # 34, (1996), Research Institute of Mathematics, Global Analysis Research Center, Seoul National University.) · Zbl 0862.49004 [14] Glowinski, R.; Lions, J.-L.; He, J., (Exact and approximate controllability for distributed parameter systems. A numerical approach, Encyclopedia of mathematics and its applications, Vol. 117, (2008), Cambridge University Press Cambridge) [15] Johnson, R.; Nerurkar, M., Controllability, stabilization, and the regulator problem for random differential systems, Memoirs of the American Mathematical Society, 136, (1998), 646 [16] Lazar, M.; Zuazua, E., Averaged control and observation of parameter-depending wave equations, Comptes Rendus de l’Academie des Sciences Paris, Série Mathématiques, 352, 497-502, (2014) · Zbl 1302.35043 [17] Lebeau, G.; Robbiano, L., Contrôle exact de l’équation de la chaleur, Communications in Partial Differential Equations, 20, 335-356, (1995) · Zbl 0819.35071 [18] Lee, E. B.; Markus, L., Foundations of optimal control theory, (1967), John Wiley & Sons, Inc. New York, London, Sydney · Zbl 0159.13201 [19] Li, J.-S., Ensemble control of finite-dimensional time-varying linear systems, IEEE Transactions on Automatic Control, 56, 2, 345-357, (2011) · Zbl 1368.93035 [20] Lions, J. L., Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. tome 1. contrôlabilité exacte, RMA, Vol. 8, (1988), Masson Paris · Zbl 0653.93002 [21] López, A.; Zuazua, E., Uniform null controllability for the one dimensional heat equation with rapidly oscillating periodic density, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 19, 5, 543-580, (2002) · Zbl 1009.35009 [22] Lu, Q.; Zuazua, E., Robust null controllability for heat equations with unknown switching control mode, Discrete and Continuous Dynamical Systems, 34, 10, 4183-4210, (2014) · Zbl 1301.93028 [23] Luca, F.; de Teresa, L., Control of coupled parabolic systems and Diophantine approximations, SeMA Journal, 61, 1, 1-17, (2013) · Zbl 1272.93029 [24] Masterkov, Yu. V.; Rodina, L. I., Controllability of a linear dynamical system with random parameters, Differential Equations, 43, 4, 469-477, (2007) · Zbl 1131.93008 [25] Micu, S.; Zuazua, E., An introduction to the controllability of linear PDE, (Sari, T., Contrôle non linéaire et applications, Collection travaux en cours, (2005), Hermann), 67-150 · Zbl 1231.93042 [26] Petersen, I. R., A notion of possible controllability for uncertain linear systems with structured uncertainty, Automatica, 45, 134-141, (2009) · Zbl 1154.93319 [27] Pytlak, R., (Numerical methods for optimal control problems with state constraints, Lecture notes in mathematics, Vol. 1707, (1999), Springer-Verlag Berlin) [28] Savkin, A. V.; Petersen, I. R., Uncertainty-averaging approach to output feedback optimal guaranteed cost control of uncertain systems, Journal of Optimization Theory and Applications, 88, 2, 321-337, (1996) · Zbl 0853.93065 [29] Seo, J.; Chung, D.; Park, C. G.; Lee, J. G., The robustness of controllability and observability for discrete linear time-varying systems with norm-bounded uncertainty, IEEE Transactions on Automatic Control, 50, 7, 1039-1043, (2005) · Zbl 1365.93046 [30] Trélat, E., (Contrôle optimal: théorie et applications, Mathématiques concrètes, (2005), Vuibert Paris) [31] Tröltzsch, F., (Optimal control of partial differential equations: theory, methods and applications, Graduate studies in mathematics, Vol. 112, (2010), American Mathematical Society Providence, RI) [32] Tucsnak, M.; Weiss, G., (Observation and control for operator semigroups, Birkhäuser advanced texts: Basler Lehrbücher, (2009), Birkhäuser Verlag Basel) [33] Zuazua, E., Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47, 2, 197-243, (2005) · Zbl 1077.65095 [34] Zuazua, E., Controllability and observability of partial differential equations: some results and open problems, (Dafermos, C. M.; Feireisl, E., Handbook of differential equations: evolutionary equations, Vol. 3, (2006), Elsevier Science Amsterdam), 527-621 · Zbl 1193.35234
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