In this paper a nonmonotone limited memory BFGS (NLBFGS) method is applied for approximately solving optimal control problems (OCPs) governed by one-dimensional parabolic partial differential equations. A discretized optimal control problem is obtained by using piecewise linear finite element and well-known backward Euler methods. Afterwards, regarding the implicit function theorem, the optimal control problem is transformed into an unconstrained nonlinear optimization problem (UNOP). Finally the obtained UNOP is solved by utilizing the NLBFGS method. In comparison to other existing methods, the NLBFGS method shows a significant improvement especially for nonlinear and ill-posed control problems.
The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques.
In this paper, we study the problem of finding deterministic (also known as feedback or closed-loop) Markov Nash equilibria for a class of discrete-time stochastic games. In order to establish our results, we develop a potential game approach based on the dynamic programming technique. The identified potential stochastic games have Borel state and action spaces and possibly unbounded nondifferentiable cost-per-stage functions. In particular, the team (or coordination) stochastic games and the stochastic games with an action independent transition law are covered.
This note investigates the optimal control problem for a time-invariant linear systems with an arbitrary constant time-delay in in the input channel. A state feedback is provided for the infinite horizon case with a quadratic cost function. The solution is memoryless, except at an initial time interval of measure equal to the time-delay. If the initial input is set equal to zero, then the optimal feedback control law is memoryless from the beginning. Stability results are established for the closed loop system, in the scalar case.
This paper is concerned with the finite and infinite horizon optimal control issue for a class of networked control systems with stochastic communication protocols. Due to the limitation of networked bandwidth, only the limited number of sensors and actuators are allowed to get access to network mediums according to stochastic access protocols. A discrete-time Markov chain with a known transition probability matrix is employed to describe the scheduling behaviors of the stochastic access protocols, and the networked systems are modeled as a Markov jump system based on the augmenting technique. In such a framework, both the approaches of stochastic analysis and dynamic programming are utilized to derive the optimal control sequences satisfying the quadratic performance index. Moreover, the optimal controller gains are characterized by solving the solutions to coupled algebraic Riccati equations. Finally, a numerical example is provided to demonstrate the correctness and effectiveness of the proposed results.
We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.
An optimization problem for the unilateral contact between a pseudoplate and a rigid obstacle is considered. The variable thickness of the pseudoplate plays the role of a control variable. The cost functional is a regular functional only in the smooth case. The existence of an optimal thickness is verified. The penalized optimal control problem is considered in the general case.
A distributed optimal control problem on and inside a homogeneous skin tissue is solved subject to Pennes' equation with Dirichlet boundary condition at one end and Rubin condition at the other end. The point heating power induced by conducting heating probe inserted at the tumour site as an unknown control function at specific depth inside biological body is preassigned. Corresponding pseudo-port Hamiltonian system is proposed. Moreover, it is proved that bioheat transfer equation forms a contraction and dissipative system. Mild solution for bioheat transfer equation and its adjoint problem are proposed. Controllability and exponentially stability for the related system is proved. The optimal control problem is solved using strongly continuous semigroup solution and time discretization. Mathematical simulations for a thermal therapy in the presence of point heating power are presented to investigate efficiency of the proposed technique.
Suppose that at any stage of a statistical experiment a control variable X that affects the distribution of the observed data Y at this stage can be used. The distribution of Y depends on some unknown parameter θ, and we consider the problem of testing multiple hypotheses H1:θ=θ1, H2:θ=θ2,…, Hk:θ=θk allowing the data to be controlled by X, in the following sequential context. The experiment starts with assigning a value X1 to the control variable and observing Y1 as a response. After some analysis, another value X2 for the control variable is chosen, and Y2 as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses H1,…, Hk is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations Y1,Y2,…,Yn are independent, given controls X1,X2,…,Xn, n=1,2,….
In this paper, the problems on purposefully controlling chaos for a three-dimensional quadratic continuous autonomous chaotic system, namely the chaotic Pehlivan-Uyaroglu system are investigated. The chaotic system, has three equilibrium points and more interestingly the equilibrium points have golden proportion values, which can generate single folded attractor. We developed an optimal control design, in order to stabilize the unstable equilibrium points of this system. Furthermore, we propose Lyapunov stability to control the Pehlivan-Uyaroglu system with unknown parameters by way of a feedback control approach and a single controller. Numerical simulations are performed to demonstrate the effectiveness of the proposed control strategies.