On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials

Abstract In this paper we prove the optimality of the observability inequality for parabolic systems with potentials in even space dimensions n ⩾ 2 . This inequality (derived by E. Fernandez-Cara and the third author in the context of the scalar heat equation with potentials in any space dimension) asserts, roughly, that for small time, the total energy of solutions can be estimated from above in terms of the energy localized in a subdomain with an observability constant of the order of exp ( C ‖ a ‖ ∞ 2 / 3 ) , a being the potential involved in the system. The problem of the optimality of the observability inequality remains open for scalar equations. The optimality is a consequence of a construction due to V.Z. Meshkov of a complex-valued bounded potential q = q ( x ) in R 2 and a nontrivial solution u of Δ u = q ( x ) u with the decay property | u ( x ) | ⩽ exp ( − | x | 4 / 3 ) . Meshkov's construction may be generalized to any even dimension. We give an extension to odd dimensions, which gives a sharp decay rate up to some logarithmic factor and yields a weaker optimality result in odd space-dimensions. We address the same problem for the wave equation. In this case it is well known that, in space-dimension n = 1 , observability holds with a sharp constant of the order of exp ( C ‖ a ‖ ∞ 1 / 2 ) . For systems in even space dimensions n ⩾ 2 we prove that the best constant one can expect is of the order of exp ( C ‖ a ‖ ∞ 2 / 3 ) for any T > 0 and any observation domain. Based on Carleman inequalities, we show that the positive counterpart is also true when T is large enough and the observation is made in a neighborhood of the boundary. As in the context of the heat equation, the optimality of this estimate is open for scalar equations. We address similar questions, for both equations, with potentials involving the first order term. We also discuss issues related with the impact of the growth rates of the nonlinearities on the controllability of semilinear equations. Some other open problems are raised.

[1]  Shmuel Agmon,et al.  Lectures on exponential decay of solutions of second order elliptic equations : bounds on eigenfunctions of N-body Schrödinger operators , 1983 .

[2]  H. Fattorini.  Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation , 1977 .

[3]  L. Hörmander.  Non-uniqueness for the Cauchy Problem , 1975 .

[4]  J. Uchiyama.  Lower bounds of decay order of eigenfunctions of second-order elliptic operators , 1985 .

[5]  Michel C. Delfour,et al.  Quantum Control - Mathematical and Numerical Challenges: CRM Workshop, Montreal, Canada, October 6-11, 2002 , 2003, CRM Workshop.

[6]  Lionel Rosier,et al.  Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain , 1997 .

[7]  Oleg Yu. Imanuvilov,et al.  On Carleman estimates for hyperbolic equations , 2002 .

[8]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[9]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[10]  T. Wolff.  A counterexample in a unique continuation problem , 1994 .

[11]  Vilmos Komornik,et al.  Controllability of semilinear wave equations with infinitely iterated logarithms , 2002 .

[12]  C. Bardos,et al.  Asymptotic decay of the solution of a second-order elliptic equation in an unbounded domain. Applications to the spectral properties of a Hamiltonian , 1977, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[14]  M. S. Baouendi,et al.  A non uniqueness result for operators of principal type , 1995 .

[15]  Xu Zhang,et al.  Exact controllability of semilinear plate equations , 2001 .

[16]  Enrique Zuazua,et al.  Concentration and Lack of Observability¶of Waves in Highly Heterogeneous Media , 2002 .

[17]  Oleg Yu. Imanuvilov,et al.  Controllability of Evolution equations , 1996 .

[18]  Luc Miller,et al.  Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time , 2003, math/0307158.

[19]  F. Colombini,et al.  UNIQUENESS AND NONUNIQUENESS IN THE CAUCHY PROBLEM FOR DEGENERATE ELLIPTIC OPERATORS , 1993 .

[20]  Enrique Zuazua,et al.  Exact controllability for semilinear wave equations in one space dimension , 1993 .

[21]  Enrique Zuazua,et al.  Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations , 2000 .

[22]  Thomas I. Seidman,et al.  The ‘window problem’ for series of complex exponentials , 2000 .

[23]  Xu Zhang,et al.  Exact Controllability for Multidimensional Semilinear Hyperbolic Equations , 2007, SIAM J. Control. Optim..

[24]  Enrique Zuazua,et al.  Problem 5.5 Exact controllability of the semi-linear wave equation , 2009 .

[25]  David L. Russell,et al.  A Unified Boundary Controllability Theory for Hyperbolic and Parabolic Partial Differential Equations , 1973 .

[26]  Enrique Zuazua,et al.  Null and approximate controllability for weakly blowing up semilinear heat equations , 2000 .

[27]  Enrique Zuazua,et al.  Remarks on the controllability of the Schrödinger equation , 2002, CRM Workshop.

[28]  Enrique Zuazua,et al.  On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient , 2002, SIAM J. Control. Optim..

[29]  Enrique Zuazua,et al.  The cost of approximate controllability for heat equations: the linear case , 2000, Advances in Differential Equations.

[30]  A Pointwise Lower Bound for Positive Solutions of a Schrödinger Equation in RN , 1997 .

[31]  V. Meshkov.  ON THE POSSIBLE RATE OF DECAY AT INFINITY OF SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS , 1992 .