Special Session 112: Controllability and Stabilization of Partial Differential Equations

Controllability for parabolic equations with large parameters.

Felipe W. Chaves-Silva Federal University of Paraiba Brazil Co-Author(s):    N. Carreno & M. G. Ferreira-Silva

Abstract: We consider a parabolic equation perturbed by an anomalous diffusion term scaled by a large parameter. We study the cost of null controllability for the system as the parameter goes to infinity and, using spectral analysis, show that is possible to construct null controls that exhibit exponential decay with respect to the parameter. Additionally, we construct uniform controls for some nonlinear versions as well.

Null Controllability of Nonlinear Wave Equations

Yan Cui Jinan University Peoples Rep of China Co-Author(s):    Peng Lu, Yi Zhou

Abstract: In this talk, we mainly introduce the controllability of several types of nonlinear wave equations. Firstly, by using the Galerkin method, we obtain the null controllability of a class of semilinear wave equations. Secondly, based on the Picard iteration method, we establish the local null controllability of a class of quasilinear equations. The talk is based on joint work with Peng Lu and Yi Zhou.

Feedback stabilization of entropy solutions to the p-system at a junction

Nicola N De Nitti EPFL Switzerland Co-Author(s):    G. M. Coclite, M. Garavello, F. Marcellini.

Abstract: We consider the p-system in Eulerian coordinates on a star-shaped network. Under suitable transmission conditions at the junction and dissipative boundary conditions in the exterior vertices, we show that the entropy solutions of the system are exponentially stabilizable. Our proof extends the strategy by Coron et al. (2017) and is based on a front-tracking algorithm used to construct approximate piecewise constant solutions whose BV norms are controlled through a suitable exponentially-weighted Glimm-type Lyapunov functional. This talk is based on a joint work with G. M. Coclite, M. Garavello, and F. Marcellini.

Minimal control time for the internal exact controllability of 1D linear hyperbolic balance laws

Long Hu Shandong University Peoples Rep of China Co-Author(s):    Guillaume Olive

Abstract: In this talk, we are concerned with the internal exact controllability of 1D linear hyperbolic balance laws when the number of controls is equal to the number of state variables. The controls are supported in space in an arbitrary nonempty open subset. The necessary and sufficient conditions will be proposed in order to characterize the minimal control time for such systems. This talk is based on a joint work with Guillaume Olive.

Disturbance rejection approaches of Korteweg-de Vries-Burgers equation under event-triggering mechanism

Wen Kang Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Wen Kang, Jing Zhang, Jun-Min Wang

Abstract: In this talk, disturbance rejection approaches are suggested to stabilize Korteweg-de Vries-Burgers (KdVB) equation under the averaged measurements. Here two approaches-active disturbance rejection control (ADRC) and disturbance observer-based control (DOBC), are introduced to reject the external unknown disturbance actively. The main challenging issue is to design the effective extended state observer (ESO)/disturbance observer (DO) for KdVB equation respectively. As for ADRC strategy, the disturbance is rejected on the basis of an ESO. As for DOBC strategy, a DO is constructed to estimate the disturbance formulated by an exogenous system. Continuous-time and event-triggered anti-disturbance controllers are further presented for distributed stabilization of KdVB system. To significantly reduce the amount of control updates, an event-triggering mechanism is utilized and the Zeno behaviour is avoided. Sufficient stability conditions are established via Lyapunov functional method. The effectiveness of proposed approaches is verified by simulation results.

Null controllability of underactuated linear parabolic-transport system

Pierre Lissy Ecole nationales des ponts et chaussees France Co-Author(s):    Armand Koenig

Abstract: I will present controllability properties of mixed systems of linear parabolic-transport equations, with possibly nondiagonalizable diffusion matrix, on the 1D torus, coupled by constant coupling terms. The distributed control acts through a constant matrix, with possibly less controls than equations. In small time or for not regular enough initial data, these systems are never controllable, whereas in large time, null-controllability holds, for regular initial data, iff a spectral Kalman rank condition is verified.

Global controllability of the Boussinesq system by using a degenerate temperature control

Vahagn Nersesyan NYU Shanghai Peoples Rep of China Co-Author(s):    Manuel Rissel

Abstract: We will prove that the 2D incompressible Boussinesq system on the torus is globally approximately controllable via physically localized control appearing only in the temperature equation. In addition, our controls have an explicitly prescribed structure; even without such structural requirements, large data controllability results for Boussinesq flows driven merely by a physically localized temperature profile were so far unknown. The presented method exploits various connections between the model`s underlying transport-, coupling-, and scaling mechanisms. This is a joint work with Manuel Rissel (NYU Shanghai).

Second-Order Necessary Conditions for Stochastic Optimal Control Problems with Final Point Constraints

Haisen Zhang School of Mathematical Science, Sichuan Normal University Peoples Rep of China Co-Author(s):

Abstract: We establish some second-order necessary conditions for optimal control problems of stochastic differential equations with final points constraints. Both the drift and the diffusion terms of the control systems can contain the control variable, and the control regions are allowed to be nonconvex. The classical variational analysis approach and the inverse mapping theorem are used to derive those second-order necessary conditions.

Stability Analysis of an Abstract System with Local Damping

Qiong Zhang Beijing Institute of Technolog Peoples Rep of China Co-Author(s):    Chenxi Deng, Otared Kavian

Abstract: We consider an abstract system of the type $u_{tt} + Lu + Bu_{t} = 0$, where $L$ is a self-adjoint operator on a Hilbert space and operator $B$ represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator $L$ and non-degeneration of operator $B$. Finally, several examples are given as application of our abstract results.

Output regulation for a 1-D wave equation with velocity recirculation and disturbances

Hua-Cheng Zhou Central South University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider the output regulation problem for a 1-D wave equation with velocity recirculation where the disturbances in all possible channels of the wave equation and the reference signal are generated from an unknown finite-dimensional exosystem. An adaptive error-based observer is proposed to effectively estimate all unknown frequencies, and based on this, a tracking-error-based feedback control is designed to regulate the output to the reference signal exponentially. This talk is based on a joint work with Fanggang Hu.