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Optimal actuator Design for semilinear systems. (English) Zbl 1420.49007

Summary: Actuator location and design are important choices in controller design for distributed parameter systems. Semilinear partial differential equations model a wide spectrum of physical systems with distributed parameters. It is shown that under certain conditions on the nonlinearity and the cost function, an optimal control input together with an optimal actuator choice exists. First-order necessary optimality conditions are derived. The results are applied to optimal actuator and controller design in a nonlinear railway track model as well as semilinear wave models.

MSC:

49J27 Existence theories for problems in abstract spaces
49K27 Optimality conditions for problems in abstract spaces
49J20 Existence theories for optimal control problems involving partial differential equations
49J50 Fréchet and Gateaux differentiability in optimization
35L71 Second-order semilinear hyperbolic equations
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[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amst.) 140, Elsevier/Academic Press, Amsterdam, 2003.
[2] M. Ansari, E. Esmailzadeh, and D. Younesian, Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads, J. Sound Vib., 330 (2011), pp. 1455-1471.
[3] C. Antoniades and P. D. Christofides, Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport-reaction processes, Chem. Eng. Sci., 56 (2001), pp. 4517-4535.
[4] A. Armaou and M. A. Demetriou, Robust detection and accommodation of incipient component and actuator faults in nonlinear distributed processes, AIChE J., 54 (2008), pp. 2651-2662.
[5] H. T. Banks and K. Ito, A unified framework for approximation in inverse problems for distributed parameter systems, Control Theory Adv. Tech., 4 (1988), pp. 73-90.
[6] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4th ed., Springer Monographs in Mathematics, Springer, Dordrecht, 2012. · Zbl 1244.49001
[7] M. Bergounioux and K. Kunisch, On the structure of Lagrange multipliers for state-constrained optimal control problems, Systems Control Lett., 48 (2003), pp. 169-176. · Zbl 1134.49310
[8] J. L. Boldrini, B. M. C. Caretta, and E. Fernández-Cara, Some optimal control problems for a two-phase field model of solidification, Rev. Mat. Complut., 23 (2009), pp. 49-75. · Zbl 1182.49020
[9] R. Buchholz, H. Engel, E. Kammann, and F. Tröltzsch, On the optimal control of the Schlögl-model, Comput. Optim. Appl., 56 (2013), pp. 153-185. · Zbl 1273.49006
[10] E. Casas, Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), pp. 1297-1327, https://doi.org/10.1137/S0363012995283637. · Zbl 0893.49017
[11] E. Casas, C. Ryll, and F. Tröltzsch, Sparse optimal control of the Schlögl and FitzHugh-Nagumo systems, Comput. Methods Appl. Math., 13 (2013), pp. 415-442. · Zbl 1393.49019
[12] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), pp. 15-55. · Zbl 0633.47025
[13] G. Ciaramella and A. Borzi, Quantum optimal control problems with a sparsity cost functional, Numer. Funct. Anal. Optim., 37 (2016), pp. 938-965. · Zbl 1404.81085
[14] T. Dahlberg, Dynamic interaction between train and nonlinear railway track model, in Proceedings of the Fifth Eurooean Conference on Structural Dynamics, Munich, Germany, 2002, pp. 1155-1160.
[15] J. C. de los Reyes, R. Herzog, and C. Meyer, Optimal control of static elastoplasticity in primal formulation, SIAM J. Control Optim., 54 (2016), pp. 3016-3039, https://doi.org/10.1137/130920861. · Zbl 1386.49029
[16] M. S. Edalatzadeh and A. Alasty, Boundary exponential stabilization of non-classical micro/nano beams subjected to nonlinear distributed forces, Appl. Math. Model., 40 (2016), pp. 2223-2241. · Zbl 1452.74062
[17] M. S. Edalatzadeh and K. A. Morris, Stability and well-posedness of a nonlinear railway track model, IEEE Control Syst. Lett., 3 (2019), pp. 162-167.
[18] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer-Verlag, New York, 2000.
[19] F. Fahroo and M. A. Demetriou, Optimal actuator/sensor location for active noise regulator and tracking control problems, J. Comput. Appl. Math., 114 (2000), pp. 137-158. · Zbl 0966.76083
[20] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia Math. Appl. 62, Cambridge University Press, Cambridge, 1999.
[21] A. Fleig and R. Guglielmi, Optimal control of the Fokker-Planck equation with space-dependent controls, J. Optim. Theory Appl., 174 (2017), pp. 408-427. · Zbl 1373.35311
[22] M. I. Frecker, Recent advances in optimization of smart structures and actuators, J. Intell. Material Syst. Struct., 14 (2003), pp. 207-216.
[23] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), pp. 185-207. · Zbl 1094.35082
[24] S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations, 13 (2008), pp. 1051-1074. · Zbl 1183.35035
[25] S. Gerbi and B. Said-Houari, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions, Nonlinear Anal., 74 (2011), pp. 7137-7150. · Zbl 1228.35150
[26] M. Hintermüller, T. Keil, and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with nonmatched fluid densities, SIAM J. Control Optim., 55 (2017), pp. 1954-1989, https://doi.org/10.1137/15M1025128. · Zbl 1368.49022
[27] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Math. Model. Theory Appl. 23, Springer, New York, 2009.
[28] D. Hömberg, C. Meyer, J. Rehberg, W. Ring, and D. H. Omberg, Optimal control for the thermistor problem, SIAM J. Control Optim., 48 (2010), pp. 3449-3481, https://doi.org/10.1137/080736259. · Zbl 1203.35285
[29] D. Kalise, K. Kunisch, and K. Sturm, Optimal actuator design based on shape calculus, Math. Models Methods Appl. Sci., 28 (2018), pp. 2667-2717. · Zbl 1411.49030
[30] D. Kasinathan and K. Morris, H_\infty-optimal actuator location · Zbl 1369.93176
[31] J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), pp. 1417-1429, https://doi.org/10.1137/0325078. · Zbl 0632.93057
[32] S.-J. Kimmerle, M. Gerdts, and R. Herzog, Optimal control of an elastic crane-trolley-load system–a case study for optimal control of coupled ODE-PDE systems, Math. Comput. Model. Dyn. Syst., 24 (2018), pp. 182-206. · Zbl 1486.49002
[33] J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations, 91 (1991), pp. 355-388. · Zbl 0802.73052
[34] G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich, Constrained Optimization and Optimal Control for Partial Differential Equations, Internat. Ser. Numer. Math. 160, Birkhäuser/Springer, Basel, 2012. · Zbl 1231.49001
[35] C. Li, E. Feng, and J. Liu, Optimal control of systems of parabolic PDEs in exploitation of oil, J. Appl. Math. Comput., 13 (2003), pp. 247-259. · Zbl 1039.49025
[36] Y. Lou and P. D. Christofides, Optimal actuator/sensor placement for nonlinear control of the Kuramoto-Sivashinsky equation, IEEE Trans. Control Syst. Technol., 11 (2003), pp. 737-745.
[37] A. Martínez, C. Rodríguez, and M. E. Vázquez-Méndez, Theoretical and numerical analysis of an optimal control problem related to wastewater treatment, SIAM J. Control Optim., 38 (2000), pp. 1534-1553, https://doi.org/10.1137/S0363012998345640. · Zbl 0961.49002
[38] J. Merger, A. Borzi, and R. Herzog, Optimal control of a system of reaction-diffusion equations modeling the wine fermentation process, Optimal Control Appl. Methods, 38 (2017), pp. 112-132. · Zbl 1356.93011
[39] C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM J. Control Optim., 55 (2017), pp. 2206-2234, https://doi.org/10.1137/15M1040426. · Zbl 1376.49004
[40] S. H. Moon, Finite element analysis and design of control system with feedback output using piezoelectric sensor/actuator for panel flutter suppression, Finite Elem. Anal. Des., 42 (2006), pp. 1071-1078.
[41] K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control, 56 (2011), pp. 113-124. · Zbl 1368.93278
[42] K. Morris and S. Yang, Comparison of actuator placement criteria for control of structures, J. Sound Vib., 353 (2015), pp. 1-18.
[43] K. A. Morris, Noise reduction achievable by point control, ASME J. Dyn. Syst. Meas. Control, 120 (1998), pp. 216-223.
[44] K. A. Morris, M. A. Demetriou, and S. D. Yang, Using H_2-control performance metrics for infinite-dimensional systems, IEEE Trans. Automat. Control, 60 (2015), pp. 450-462. · Zbl 1360.93318
[45] K. A. Morris and A. Vest, Design of damping for optimal energy dissipation of vibrations, in Proceedings of the IEEE 55th Conference on Decision and Control, 2016, pp. 532-536.
[46] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[47] Y. Privat, E. Trélat, and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), pp. 1097-1126. · Zbl 1296.49004
[48] Y. Privat, E. Trélat, and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), pp. 514-544. · Zbl 1302.93052
[49] J. P. Raymond and H. Zidani, Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations, Appl. Math. Optim., 39 (1999), pp. 143-177. · Zbl 0922.49013
[50] M. R. Saviz, An optimal approach to active damping of nonlinear vibrations in composite plates using piezoelectric patches, Smart Mater. Struct., 24 (2015), 115024.
[51] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci. 143, Springer-Verlag, New York, 2002.
[52] J. Simon, Compact sets in the space L^p (0, T; B), Ann. Mat. Pura Appl. (4), 146 (1986), pp. 65-96.
[53] M. Sprengel, G. Ciaramella, and A. Borz\`\i, Investigation of optimal control problems governed by a time-dependent Kohn-Sham model, J. Dyn. Control Syst., 24 (2018), pp. 657-679. · Zbl 1407.35170
[54] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, Grad. Stud. Math. 112, American Mathematical Society, Providence, RI, 2010. · Zbl 1195.49001
[55] A. Unger and F. Tröltzsch, Fast solution of optimal control problems in the selective cooling of steel, ZAMM Z. Angew. Math. Mech., 81 (2001), pp. 447-456. · Zbl 0993.49024
[56] M. Van De Wal and B. De Jager, A review of methods for input/output selection, Automatica J. IFAC, 37 (2001), pp. 487-510. · Zbl 0995.93002
[57] A. Wouk, A Course of Applied Functional Analysis, John Wiley & Sons, New York, 1979. · Zbl 0407.46001
[58] X. Wu, B. Jacob, and H. Elbern, Optimal control and observation locations for time-varying systems on a finite-time horizon, SIAM J. Control Optim., 54 (2015), pp. 291-316, https://doi.org/10.1137/15M1014759. · Zbl 1339.93127
[59] I. Yousept, Optimal control of non-smooth hyperbolic evolution Maxwell equations in type-II superconductivity, SIAM J. Control Optim., 55 (2017), pp. 2305-2332, https://doi.org/10.1137/16M1074229. · Zbl 1377.35236
[60] A. Zettl, Sturm-Liouville Theory, Math. Surveys Monogr. 121, American Mathematical Society, Providence, RI, 2005.
[61] M. Zhang and K. A. Morris, Sensor choice for minimum error variance estimation, IEEE Trans. Automat. Control, 63 (2018), pp. 315-330. · Zbl 1390.93412
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