## Constrained approximate null controllability of the coupled heat equation with impulse controls.(English)Zbl 1479.35522

The authors discuss an issue of the constrained approximate null controllability of a systems of heat equations coupled with a real matrix $$P$$. The controls acting on the system are impulsive and periodical having the form of a series of real matrices $$Q$$. It is supposed that the controls are bounded by instant constraints. Two interpretations of the considered controlled system are given. The authors give sufficient conditions of global constrained approximate null controllability in the cases where the control acts on the system globally or locally. These conditions impose certain requirements on matrices $$P$$ and $$Q$$. In the case where the controls act globally, necessary conditions of such controllability are also determined.

### MSC:

 35K51 Initial-boundary value problems for second-order parabolic systems 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 35K90 Abstract parabolic equations 47D06 One-parameter semigroups and linear evolution equations 35R12 Impulsive partial differential equations
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### References:

 [1] N. U. Ahmed, Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints, J. Optim. Theory Appl., 47 (1985), pp. 129-158. · Zbl 0549.49028 [2] F. Ammar-Khodja, A. Benabdallah, C. Dupaix, and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., 1 (2009), pp. 427-457. · Zbl 1170.93008 [3] F. Ammar-Khodja, A. Benabdallah, and M. González-Burgos, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), pp. 267-306. [4] K. Le Balc’h, Null-controllability of two species reaction-diffusion system with nonlinear coupling: A new duality method, SIAM J. Control Optim., 57 (2019), pp. 2541-2573. · Zbl 1461.35124 [5] B. R. Barmish and W. E. Schmitendorf, A necessary and sufficient condition for local constrained controllability of a linear system, IEEE Trans. Automat. Control, 25 (1980), pp. 97-100. · Zbl 0426.93012 [6] D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), pp. 125-135. · Zbl 1039.35013 [7] O. Cârjă, Constraint controllability for linear control systems, Ann. Mat. Pura Appl., 158 (1991), pp. 13-32. · Zbl 0749.93010 [8] J.-M. Coron and J.-P. Guilleron, Control of three heat equations coupled with two cubic nonlinearities, SIAM J. Control Optim., 55 (2017), pp. 989-1019. · Zbl 1432.35113 [9] Y. Duan, L. Wang, and C. Zhang, Minimal time impulse control of an evolution equation, J. Optim. Theory Appl., 183 (2019), pp. 902-919. · Zbl 1430.49035 [10] Y. Duan and L. Wang, Minimal norm control problem governed by semilinear heat equation with impulse control, J. Optim. Theory Appl., 184 (2020), pp. 400-418. · Zbl 1434.49029 [11] M. Duprez, Controllability of a 2 x 2 parabolic system by one force with space-dependent coupling term of order one, ESAIM Control Optim. Calc. Var., 23 (2017), pp. 1473-1498. · Zbl 1375.93018 [12] P. Érdi and J. Tóth, Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models, Princeton University Press, Princeton, NJ, 1989. · Zbl 0696.92027 [13] L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture Notes, University of California, Department of Mathematics, Berkeley, CA, 2005. [14] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 2010. · Zbl 1194.35001 [15] S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim., 46 (2007), pp. 379-394. · Zbl 1146.93011 [16] T. Hillen and K. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), pp. 183-217. · Zbl 1161.92003 [17] D. Lauffenburger, R. Aris, and K. Keller, Effects of cell motility and chemotaxis on microbial population growth, Biophys. J., 40 (1982), pp. 209-219. [18] P. Lissy and E. Zuazua, Internal observability for coupled systems of linear partial differential equations, SIAM J. Control Optim., 57 (2019), pp. 832-853. · Zbl 1409.35059 [19] K. Narukawa, Admissible null controllability and optimal time control, Hiroshima Math. J., 11 (1981), pp. 533-551. · Zbl 0494.49028 [20] K. Narukawa, Admissible controllability of vibrating systems with constrained controls, SIAM J. Control Optim., 20 (1982), pp. 770-782. · Zbl 0511.93042 [21] L. Pandolfi, Linear control systems: Controllability with constrained controls, J. Optim. Theory Appl., 19 (1976), pp. 577-585. · Zbl 0309.93002 [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. · Zbl 0516.47023 [23] G. Peichl and W. Schappacher, Constrained controllability in Banach spaces, SIAM J. Control Optim., 24 (1986), pp. 1261-1275. · Zbl 0612.49026 [24] K. D. Phung, L. Wang, and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), pp. 477-499. · Zbl 1295.49005 [25] S. Qin and G. Wang, Controllability of impulse controlled systems of heat equations coupled by constant matrices, J. Differential Equations, 263 (2017), pp. 6456-6493. · Zbl 1370.93053 [26] S. Qin, G. Wang, and H. Yu, Stabilization on periodic impulse control systems, SIAM J. Control Optim., 59 (2021), pp. 1136-1160. · Zbl 1458.93121 [27] W. E. Schmitendorf and B. R. Barmish, Null controllability of linear systems with constrained controls, SIAM J. Control Optim., 18 (1980), pp. 327-345. · Zbl 0457.93012 [28] N. Shigesada, K. Kawasaki, and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), pp. 83-99. [29] N. K. Son and N. Van Su, Linear periodic control systems: Controllability with restrained controls, Appl. Math. Optim., 14 (1986), pp. 173-185. · Zbl 0616.93010 [30] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer Science and Business Media, New York, 2013. [31] E. Trélat, L. Wang, and Y. Zhang, Impulse and sampled-data optimal control of heat equations, and error estimates, SIAM J. Control Optim., 54 (2016), pp. 2787-2819. · Zbl 1370.35158
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