Norm Optimal Control Problems Research Articles

Time and norm optimal controls for the heat equation with dynamical boundary conditions

In this paper, we deal with time and norm optimal control problems for the heat equation with dynamic boundary conditions. We prove the existence of admissible controls and, combining a suitable Carleman estimate for such an equation with some analyticity results, we show a strong unique continuation property. Then, with the aid of this unique continuation, bang‐bang properties and the connection between time and norm optimal controls can be established.

Exact controllability of linear mean‐field stochastic systems and observability inequality for mean‐field backward stochastic differential equations

AbstractThis paper is concerned with the exact controllability of linear mean‐field stochastic systems with time‐variant random coefficients. We prove that the exact controllability, the validity of the observability inequality for the dual equation, the unique solvability of a family of optimal control problems, the unique solvability of a family of mean‐field forward‐backward stochastic differential equations (MF‐FBSDEs), and the unique solvability of a family of norm optimal control problems are all equivalent. Therefore, some approaches are provided to investigate the issue of exact controllability.

Minimum Time and Minimum Energy for Linear Systems; a Variational Approach

In this paper we develop a variational approach for the norm optimal control problem in an abstract general setting for linear systems. This technique is also related to the classical pseudoinverse for linear and bounded operators between Hilbert spaces. Characterizations for time optimal controls are also given.

Minimum time and minimum energy for linear control systems

In this paper we give estimates concerning regularity properties of the minimum time function and the minimum norm controls involving the controllability function given by the observability inequalities. We further study the continuity of the minimum time function in two variables (the initial point and the radius of the ball that limits the control). We prove a maximum principle for both time and norm optimal control problems.

On the partial controllability of SDEs and the exact controllability of FBSDES

A notion of partial controllability (also can be called directional controllability or output controllability) is proposed for linear controlled (forward) stochastic differential equations (SDEs), which characterizes the ability of the state to reach some given random hyperplane. It generalizes the classical notion of exact controllability. For time-invariant system, checkable rank conditions ensuring SDEs’ partial controllability are provided. With some special setting, the partial controllability for SDEs is proved to be equivalent to the exact controllability for linear controlled forward-backward stochastic differential equations (FBSDEs). Moreover, we obtain some equivalent conclusions to partial controllability for SDEs or exact controllability for FBSDEs, including the validity of observability inequalities for the adjoint equations, the solvability of some optimal control problems, the solvability of norm optimal control problems, and the non-singularity of a random version of Gramian matrix.

Minimal Norm Control Problem Governed by Semilinear Heat Equation with Impulse Control

This paper studies a kind of minimal norm optimal control problem for a semilinear heat equation with impulse controls; it consists in finding an impulse control, which has the minimal norm among all controls steering solutions of the controlled system to a given target from an initial state in a fixed time interval. We will study the existence of optimal controls to this problem. Then, we will establish a nontrivial Pontryagin’s maximum principle of this problem.

The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian

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 ℝd. 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.

Equivalence between minimal time control problems and minimal norm control problems

This paper presents some of our new developments about the equivalence between time optimal control problems and norm optimal control problems. Besides, several applications of this equivalence are given.

Some optimal control problems of heat equations with weighted controls

In this paper, the time and norm optimal control problems of controlled heat equations with a weight function are considered. For the time optimal problems, we study the following two cases: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null controllability. We prove the solvability, and obtain the bang-bang principle of the time optimal controls for aforementioned both cases. For the norm optimal control problems, we focus on equations with multi-time and multi-domain control, and present the solvability of these problems.

A Convex Characterization of Multidimensional Linear Systems Subject to SQI Constraints

This technical note proposes a convex characterization of the set of all stable closed-loop linear systems that are obtained from a given plant, which may be multidimensional, by interconnecting it in feedback with controllers that satisfy a certain pre-selected constraint. We take an approach that is particularly useful when a doubly-coprime factorization of the plant is difficult to obtain, and a stable stabilizing controller may not exist (the plant is not strongly-stabilizable) or when one may be difficult to find; most related work requires one of these. We adopt the so-called coordinate-free approach, which, unlike Youla's parametrization, does not rely on a doubly-coprime factorization of the plant. We show that if constraints which satisfy a condition called strong quadratic invariance (SQI) are imposed on the controllers then the set of all stable closed-loop multidimensional linear systems has a convex representation, and norm-optimal control problems can be cast in convex form. Although the SQI condition is in general slightly stronger than quadratic invariance (QI), which was developed in related work, they are equivalent for common classes of problems arising in decentralized control.

Computation of Time Optimal Control Problems Governed by Linear Ordinary Differential Equations

In this paper, a novel numerical algorithm is presented to compute the optimal time of a time optimal control problem where the governing system is a linear ordinary differential equation. By the equivalence between time optimal control problem and norm optimal control problem, computation of the optimal time can be obtained by solving a sequence of norm optimal control problems, which are transferred into their Lagrangian dual problems. The nonsmooth structure of the dual problem is approximated by the iteratively reweighted least square strategy. Several numerical tests are given to show the efficiency of the proposed algorithm.

Exact controllability of linear stochastic differential equations and related problems

A notion of $L^p$ -exact controllability is introduced for linear controlled (forward) stochastic differential equations with random coefficients. Several sufficient conditions are established for such kind of exact controllability. Further, it is proved that the $L^p$ -exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$ -type norm optimal control problem are all equivalent.

Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

In this work, we utilize the existing Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, together with the telescoping series method, to establish an observability inequality from measurable subsets in time‐space variable for the parabolic equation with Grushin operator in some multidimension domains. We can apply this observability inequality to show the bang–bang property for both time optimal and norm optimal control problems for this kind of singular parabolic equation. Copyright © 2016 John Wiley & Sons, Ltd.

Perturbations of Time Optimal Control Problems for a Class of Abstract Parabolic Systems

In this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge, in an appropriate sense, to a given strictly negative operator. Our main application to PDEs systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients, as we encounter in homogenization theory. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the solutions of the time optimal control problems for the systems with oscillating coefficients converge, in the usual norms, to the solution of the corresponding problem for the homogenized system. In order to prove our main theorem, we provide several new results, which could be of a broader interest, on time and norm optimal control problems.

On optimal location of diffusion and related optimal control for null controllable heat equation

We consider optimal locations of heat diffusion and related optimal control to achieve null controllability for multi-dimensional heat equations. Both time optimal control and norm optimal control problems are considered. The reason behind combining these two problems together is that these two problems are actually equivalent: The energy to be used to drive the system to zero in minimal time interval is actually the minimal energy of driving the system to zero in this minimal time interval and visa versa. We formulate the optimal locations for time optimal control and norm optimal control into two types of shape optimization problems. One is seeking the optimal domain of heat diffusion with a fixed interior actuator domain. This can be considered as a domain perturbation problem in shape optimization. Another is to seek both the optimal locations of the optimal heat diffusion domain and the related optimal actuator domain. The existences of these two types of shape optimization problems over some class of open sets in general RN space have been proved separately.

Continuous dependence of optimal control to controlled domain of actuator for heat equation

In this paper, we consider continuous dependence of the optimal control with respect to the actuator domain which is varying as open subset in the spatial domain for a multi-dimensional heat equation. Both time optimal control and norm optimal control problems are considered. The reason behind combining these two problems together is that these two problems are actually equivalent: The energy to be used to drive the system to target set in minimal time interval is actually the minimal energy of driving the system to target set in this minimal time interval, and visa versa. It is shown that both optimal control and optimal cost are continuous with respect to open controlled actuator domain under the Lebesgue measure.

The Norm Optimal Control Problem for Stochastic Linear Control Systems

In this paper we are concerned with two norm optimal control problems for different stochastic linear control systems. One is for approximately controllable systems with the natural filtra- tion, while another is for exactly controllable systems with a general filtration. For each aforementioned norm optimal control problem, we construct the unique norm optimal control, through building up some suitable quadratic functional and making use of a variational characterization on its minimizer.

Time optimal control of the heat equation with pointwise control constraints

Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained.

The Time Optimal Control with Constraints of the Rectangular Type for Linear Time-Varying ODEs

In this paper, we study a time optimal control problem of some linear time-varying ordinary differential equations, where the control constraint set is of the rectangular type. We derive a necessary and sufficient condition and provide an algorithm for the optimal time, as well as the optimal control for this problem. The basis consists of an equivalence theorem between the time optimal control problem and an associated norm optimal control problem, and the variational characterization of the latter problem.

An observability estimate for parabolic equations from a measurable set in time and its applications

This paper presents a new observability estimate for parabolic equations in \Omega\times\left( 0,T\right) , where \Omega is a convex domain. The observation region is restricted over a product set of an open nonempty subset of \Omega and a subset of positive measure in \left( 0,T\right) . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.