# Control of PDE and Inverse Problems

**Control of PDE**

From a mathematical point of view, a control system is a dynamical system on which we can act by means of a control. This system is described by a state variable, and the aim of introducing a control is forcing the system to have a prescribed behavior. For instance, we can try to drive the system from an initial state to a final one, either in a final time or asymptotically as the time goes to infinity. Or we can look for a control minimizing some quantity given by a functional depending on the evolution of the control system.

The properties of a control system given by a PDE depend on the underlying equation. In this way, we consider, for instance, parabolic, hyperbolic or dispersive systems, and linear or nonlinear systems.

The main topics are controllability (exact, to zero or approximately), stabilization and optimal ontrol. They are studied from both theoretical (existence of controls, characterization, etc.) and numerical point of view (discretization, convergence analysis, implementation, etc.).

**Inverse problems**

An inverse problem is a task where the values of some model parameters must be obtained from the observed data resulting from the model. For instance, finding some properties of the medium from knowledge of some fields interacting with the medium. Inverse problems are natural for several applications. Some examples are Geophysical Prospecting, where given a signal and an unknown obstacle, the problem is to find what does the obstacle look like; and Medical Imaging, where the objective is to find out something about the inside of a body from non-invasive measurements.

We are mainly interested in the mathematical theory of inverse problems where the underlying phenomena is modeled by a Partial Differential Equation (PDE). In this case, an inverse problem is usually stated in this way: given some information about the solutions to a PDE, find the unknown coefficients in the governing equation. Depending on the specific problem, we are concerned with the uniqueness, stabilization and reconstruction of the unknown parameter.

## Members

## Publications

- Carreño N, Guerrero S. Uniform null controllability of a linear KdV equation using two controls.
*J. Math. Anal. Appl.*2018;457(1):922-943. [Digital version] [Bibtex] - Carreño N. Insensitizing controls for the Boussinesq system with no control on the temperature equation.
*Adv. Differential Equations*2017;22(3-4):235-258. [Digital version] [Bibtex] - Carreño N, Guzmán P. On the cost of null controllability of a linear fourth-order parabolic equation.
*J. Differential Equations*2016;261(11):6485-6520. [Digital version] [Bibtex] - Carreño N, Cerpa E. Local controllability of the stabilized Kuramoto–Sivashinsky system by a single control acting on the heat equation.
*J. Math. Pures Appl.*2016;106(4):670-694. [Digital version] [Bibtex] - Carreño N, Cerpa E, Calsavara B. Insensitizing controls for a phase field system.
*Nonlinear Anal.*2016;143:120-137. [Digital version] [Bibtex] - Araruna F. D, Cerpa E, Mercado A, Santos M. C. Internal null controllability of a linear Schrodinger-KdV system on a bounded interval.
*Journal of Differential Equations*2016;260(1):653–687. [Digital version] [Bibtex] - Carreño N, Guerrero S. On the non-uniform null controllability of a linear KdV equation.
*Asymptot. Anal.*2015;94(1-2):33-69. [Digital version] [Bibtex] - Cerpa E, Mercado A, Pazoto A. Null controllability of the stabilized Kuramoto-Sivashinsky system with one distributed control.
*SIAM J. Control Optim.*2015;53(3):1543-1568. [Digital version] [Bibtex] - Carreño N, Guerrero S, Gueye M. Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system.
*ESAIM Control Optim. Calc. Var.*2015;21(1):73-100. [Digital version] [Bibtex] - Carreño N, Gueye M. Insensitizing controls with one vanishing component for the Navier-Stokes system.
*J. Math. Pures Appl. (9)*2014;101(1):27-53. [Digital version] [Bibtex] - Baudouin L, Cerpa E, Emmanuelle C, Mercado A. On the determination of the principal coefficient from boundary measurements in a KdV equation.
*J. Inverse Ill-Posed Probl.*2014;22(6):819-846. [Digital version] [Bibtex] - Cerpa E. Control of a Korteweg-de Vries equation: a tutorial.
*Math. Control Relat. Fields*2014;4(1):45-99. [Digital version] [Bibtex] - Carreño N, Guerrero S. Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain.
*J. Math. Fluid Mech.*2013;15(1):139-153. [Digital version] [Bibtex] - Baudouin L, Cerpa E, Crépeau E, Mercado A. Lipschitz stability in an inverse problem for the Kuramoto-Sivashinsky equation.
*Appl. Anal.*2013;92(10):2084-2102. [Digital version] [Bibtex] - Bakri L, Casteras J.-B. Quantitative uniqueness for Schroedinger operator with regular potentials..
*Mathemathical methods in applied science*2013. [Digital version] [Bibtex] - Bakri L. Carleman estimate for the Schrödinger operator. Applications to quantitative uniqueness..
*Communications in Partial Differential equations*2013;38(1):69-91. [Digital version] [Bibtex] - Cerpa E, Rivas I, Zhang B.-Y. Boundary controllability of the Korteweg-de Vries equation on a bounded domain.
*SIAM J. Control Optim.*2013;51(4):2976-3010. [Digital version] [Bibtex] - Cerpa E, Coron J.-M. Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition.
*IEEE Transactions on Automatic Control*2013;58(7):1688-1695. [Digital version] [Bibtex] - Carreño N. Local controllability of the N-dimensional Boussinesq system with N−1 scalar controls in an arbitrary control domain.
*Math. Control Relat. Fields*2012;2(4):361-382. [Digital version] [Bibtex] - Bakri L. Quantitative uniqueness for Schroedinger operator.
*Indiana University Mathematics Journal*2012;61(4):1565-1580. [Digital version] [Bibtex] - Bakri L. Critical sets of eigenfunctions of the Laplacian.
*Journal of Geometry and Physics*2012;62(10):2024-2037. [Digital version] [Bibtex] - Cerpa E, Mercado A, Pazoto A. On the boundary control of a parabolic system coupling KS-KdV and Heat equations.
*Scientia Series A: Mathematical Sciences*2012;22:55-74. [Digital version] [Bibtex] - Cerpa E, Pazoto A. A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations".
*Comm. Contemp. Math.*2011;13(1):183-189. [Digital version] [Bibtex] - Cerpa E, Mercado A. Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation.
*J. Differential Equations*2011;250(4):2024-2044. [Digital version] [Bibtex] - Cerpa E. Null controllability and stabilization of a linear Kuramoto-Sivashinsky equation.
*Commun. Pure Appl. Anal.*2010;9(1):91-102. [Digital version] [Bibtex] - Smyshlyaev A, Cerpa E, Krstic M. Boundary stabilization of a 1-D wave equation with in-domain anti-damping.
*SIAM J. Control Optim.*2010;48(6):4014-4031. [Digital version] [Bibtex] - Cerpa E, Crépeau E. Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain.
*Ann. Inst. H. Poincaré Anal. Non Linéaire*2009;26(2):457-475. [Digital version] [Bibtex] - Cerpa E, Crépeau E. Rapid exponential stabilization for a linear Korteweg-de Vries equation.
*Discrete Contin. Dyn. Syst. Ser. B*2009;11(3):655-668. [Digital version] [Bibtex] - Mercado A, Osses, Rosier. Carleman inequalities and inverse problems for the Schrodinger equation.
*C. R. Math. Acad. Sci. Paris*2008;346(1-2):53-58. [Digital version] [Bibtex] - Mercado A, Osses, Rosier. Inverse problems for the Schrodinger equation via Carleman inequalities with degenerate weights.
*Inverse Problems*2008;24:15-17. [Digital version] [Bibtex] - Baudouin, Mercado A. An inverse problem for Schrodinger equations with discontinuous main coefficient.
*Appl. Anal*2008;87:1145-1165. [Digital version] [Bibtex] - Baudouin, Mercado A, Osses. A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem.
*Inverse Problems*2007;23:257-278. [Digital version] [Bibtex] - Guerrero, Mercado A, Osses. An inverse inequality for some transport-diffusion equation. Application to the regional approximate controllability.
*Asymptotic Analysis*2007;52(3-4):243-257. [Digital version] [Bibtex] - Cerpa E. Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain.
*SIAM J. Control Optim.*2007;46(3):877-899. [Digital version] [Bibtex]

## Projects

- FONDECYT postdoctorado 3150089: Control and inverse problems of partial differential equations
- Math-Amsud Control Systems and Identification Problemst COSIP 14MATH-03.
- FONDECYT Regular: CONTROLLABILITY AND INVERSE PROBLEMS FOR SCHRODINGER EQUATIONS.
- Fondecyt No. 11090161: Controllability of nonlinear partial differential equations
- MathAmsud 08MATH04: Controllability and Inverse Problems in PDE's