The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 7

In this paper, we establish the bang-bang property of time and norm optimal control problems for parabolic equations governed by time-varying fractional Laplacian, evolved in a bounded domain of ℝ$$. We firstly get a quantitative unique continuation at one point in time for parabolic equations governed by time-varying fractional Laplacian. Then, we establish an observability inequality from measurable sets in time for solutions of the above-mentioned equations. Finally, with the aid of the observability inequality, the bang-bang property of time and norm optimal control problems can be obtained.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017075
Keywords: Time optimal control, norm optimal control, bang-bang property, observability estimates, measurable sets, fractional Laplacian

Yu, Xin 1 ; Zhang, Liang 1

1 
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     title = {The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional {Laplacian}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2019},
     publisher = {EDP Sciences},
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     doi = {10.1051/cocv/2017075},
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Yu, Xin; Zhang, Liang. The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 7. doi: 10.1051/cocv/2017075

[1] J. Apraiz, L. Escauriaza, G. Wang and C. Zhang, Observability inequalities and measurable sets. J. Eur. Math. Soc. 16 (2014) 2433–2475. | Zbl | MR | DOI

[2] C. Bardos and K.D. Phung, Observation estimate for kinetic transport equations by diffusion approximation. C. R. Math. Acad. Sci. Paris 355 (2017) 640–664. | Zbl | MR | DOI

[3] L. Escauriaza, S. Montaner and C. Zhang, Observability from measurable sets for parabolic analytic evolutions and applications. J. Math. Pures Appl. 104 (2015) 837–867. | Zbl | MR | DOI

[4] H.O. Fattorini, Time optimal control of solutions of operational differential equations. J. SIAM Control 2 (1964) 54–59. | Zbl | MR

[5] H.O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems. Elsevier (2005). | Zbl | MR

[6] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Fractional differential equations: an emergent field in applied and mathematical science, in Factorisation, Singular Operators and Relative Problems. Edited by S. Samko, A. Lebre and A.F. Dos Santos. Kluwer, Boston (2002) 151–173. | Zbl | MR

[7] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier (2006). | Zbl | MR

[8] K. Kunisch and L. Wang, Time optimal controls for the linear Fitzhugh-Nagumo equation with pointwise control constraints. J. Math. Anal. Appl. 395 (2012) 114–130. | Zbl | MR | DOI

[9] K. Kunisch and L. Wang, Time optimal control of the heat equation with pointwise control constraints. ESAIM: COCV 19 (2013) 460–485. | Zbl | MR | Numdam

[10] G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Comm. Partial Diff. Eq. 20 (1995) 335–356. | Zbl | MR | DOI

[11] Q. Lü, Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta Math. Sinica, Engl. Ser. 26 (2010) 2377–2386. | Zbl | MR | DOI

[12] Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE. Math. Control Signals Syst. 28 (2016) 1–21. | Zbl | MR

[13] R. Metzler and J. Klafter, The random walk guide’s to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1–77. | Zbl | MR | DOI

[14] R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004) 161–208. | Zbl | MR | DOI

[15] S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44 (2006) 1950–1972. | Zbl | MR | DOI

[16] L. Miller, On the controllability of anomalous diffusions generated by the fractional laplacian. Math. Control Signals Syst. 18 (2006) 260–271. | Zbl | MR | DOI

[17] V.J. Mizel and T.I. Seidman, An abstract “bang-bang principle” and time-optimal boundary control of the heat equation. SIAM J. Control Optim. 35 (1997) 1204–1216. | Zbl | MR | DOI

[18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer (1983). | Zbl | MR | DOI

[19] K.D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain. J. Funct. Anal. 259 (2010) 1230–1247. | Zbl | MR | DOI

[20] K.D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications. J. Eur. Math. Soc. 15 (2013) 681–703. | Zbl | MR | DOI

[21] K.D. Phung, L.J. Wang and C. Zhang. Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. Henri Poincaré Anal. Non Lin. 31 (2014) 477–499. | Zbl | MR | Numdam | DOI

[22] I. Sokolov, J. Klafter and A. Blumen, Fractional kinetics. Phys. Today 55 (2002) 48–54. | DOI

[23] V.E. Tarasov, Fractional Dynamics. Springer (2010). | Zbl | MR | DOI

[24] G. Wang, L∞-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47 (2008) 1701–1720. | Zbl | MR | DOI

[25] G. Wang and Y. Xu, Equivalence of three different kinds of optimal control problems for heat equations and its applications. SIAM J. Control Optim. 51 (2013) 848–880. | Zbl | MR | DOI

[26] G. Wang and C. Zhang. Observability estimate from measurable sets in time for some evolution equations. SIAM J. Control Optim. 55 (2017) 1862–1886. | Zbl | MR | DOI

[27] G. Wang and Y. Zhang, Decompositions and bang-bang properties. Math. Control Relat. Fields 7 (2017) 73–170. | Zbl | MR | DOI

[28] G. Wang, Y. Xu and Y. Zhang, Attainable subspaces and the bang-bang property of time optimal controls for heat equations. SIAM J. Control Optim. 53 (2015) 592–621. | Zbl | MR | DOI

[29] C. Zhang, The time optimal control with constraints of the rectangular type for time-varying ODEs. SIAM J. Control Optim. 51 (2013) 1528–1542. | Zbl | MR | DOI

[30] Y. Zhang, Two equivalence theorems of different kinds of optimal control problems for Schrödinger equations. SIAM J. Control Optim. 53 (2015) 926–947. | Zbl | MR | DOI

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