Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support

We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports

Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional torus T = R/(2{\pi}Z). We prove a Unique Continuation Property (UCP) for small data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation. Applications to the stabilization of the above equations are given. In particular, we show that when an internal control acting on a moving interval is applied in BBM equation, then a semiglobal exponential stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H s (T) for any s \geq 1

Controllability of evolution equations with memory

First Published in SIAM Journal on Control and Optimization in Volume 55, Issue 4, 2017, Pages 2437-2459, published by the Society for Industrial and Applied Mathematics (SIAM)This article is devoted to studying the null controllability of evolution equations with memory terms. The problem is challenging not only because the state equation contains memory terms but also because the classical controllability requirement at the final time has to be reinforced, involving the contribution of the memory term, to ensure that the solution reaches the equilibrium. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the adjoint system. We first consider finite-dimensional dynamical systems involving memory terms and derive rank conditions for controllability. Then the null controllability property is established for some parabolic equations with memory terms, by means of Carleman estimatesF. W. Chaves-Silva was partially supported by the ERC project Semi- Classical Analysis of Partial Di erential Equations, ERC-2012-ADG, project number: 320845. X. Zhang was supported by the NSF of China under grant 11231007 and the Chang Jiang Scholars Program from the Chinese Education Ministry. This work was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, ICON of the French ANR (ANR-2016-ACHN-0014-01), FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of the EOARD-AFOSR, and the MTM2014-52347 Grant of the MINECO (Spain

Null controllability of the structurally damped wave equation with moving point control

International audienceWe investigate the internal controllability of the wave equation with structural damping on the one-dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions

Stabilizability for nonautonomous linear parabolic equations with actuators as distributions

The stabilizability of a general class of abstract parabolic-like equations is investigated, with a finite number of actuators. This class includes the case of actuators given as delta distributions located at given points in the spatial domain of concrete parabolic equations. A stabilizing feedback control operator is constructed and given in explicit form. Then, an associated optimal control is considered and the corresponding Riccati feedback is investigated. Results of simulations are presented showing the stabilizing performance of both explicit and Riccati feedbacks.Comment: 7 figure

About the controlability of some equations in Cardiology, Biology, Fluid Mechanics, and Viscoelasticity

In this thesis we analyze the properties of controllability and observability for selected partial differential equations which model various phenomena in cardiology, biology, fluid mechanics and viscoelasticity. We begin, in chapter 2, with the analysis of the uniform controllability of families of linear coupled parabolic systems approximating parabolic-elliptic systems. We prove, under appropriate assumptions on the coupling terms, the uniform, with respect to the degenerating parameter, null controllability of the family when only one control is acting on the system. In chapter 3, we analyze the uniform null controllability of a family of nonlinear reaction-diffusion systems approximating a nonlinear parabolic-elliptic system model- ing electrical activity in the cardiac tissue. Combining Carleman estimates and energy inequalities, we prove the uniform null controllability of the family by means of a single control. Chapter 4 studies the controllability of the parabolic Keller-Segel system of chemo- taxis which converges to its parabolic-elliptic version. We show that this nonlinear coupled parabolic system is locally uniformly controllable around a solution of the parabolic-elliptic system when the control is acting on the chemical component. In chapter 5, we consider the wave equation with both a viscous Kelvin-Voigt and a frictional damping as a model of viscoelasticity. Decomposing the system in its parabolic and hyperbolic parts, we prove the null controllability of the system when the control region, driven by the flow of an ODE, covers the whole domain. Finally, in chapter 6, we study the cost of controlling the Stokes system to zero. Using a new controllability result for a hyperbolic system with a pressure term and the control transmutation method, we show that the cost of driving the Stokes system to rest at a time $T >0$ is of order $e^{C/T}$ when $T \to 0^+$, as in the case of the heat equation

Estimation and Control of Robotic Radiation-Based Processes

This dissertation presents a closed-loop control and state estimation framework for a class of distributed-parameter processes employing a moving radiant actuator. These radiation-based processes have the potential to significantly reduce the energy consumption and environmental impact of traditional industrial processes. Successful implementation of these approaches in large-scale applications requires precise control systems. This dissertation provides a comprehensive framework for: 1) integration of trajectory generation and feedback control, 2) online distributed state and parameter estimation, and 3) optimal coordination of multiple manipulated variables, so as to achieve elaborate control of these radiation-based processes for improved process quality and energy efficiency. The developed framework addresses important issues for estimation and control of processes employing a moving radiant actuator from both practical and theoretical aspects. For practical systems, an integrated motion and process control approach is first developed to compensate for disturbances by adjusting either the radiant power of the actuator or the speed of the robot end effector based on available process measurements, such as temperature distribution. The control problem is then generalized by using a 1D scanning formulation that describes common characteristics of typical radiant source actuated processes. Based on this 1D scanning formulation, a distributed state and parameter estimation scheme that incorporates a dual extended Kalman filter (DEKF) approach is developed to provide real-time process estimation. In this estimation scheme, an activating policy accompanying the moving actuator is applied in order to reduce the computational cost and compensate for observability changes caused by the actuator\u27s movement. To achieve further improvements in process quality, a static optimization and a rule-based feedback control strategy are used to coordinate multiple manipulated variables in open-loop and closed-loop manners. Finally, a distributed model predictive control (MPC) framework is developed to integrate process optimization and closed-loop coordination of manipulated variables. Simulation studies conducted on a robotic ultraviolet (UV) paint curing process show that the developed estimation and control framework for radiant source actuated processes provide improved process quality and energy efficiency by adaptively compensating for disturbances and optimally coordinating multiple manipulated variables