Optimal Control Problem In Terms Research Articles

Optimal control of microbial production in the chemostat

In this work, we study the problem of maximizing microbial production within a continuous stirred-tank reactor with n ≥ 1 species and one substrate. The interaction between species is described by the chemostat system including a mutation factor. The optimization problem falls into an optimal control problem of Lagrange type in which the control parameter is the dilution rate of the reactor. Thanks to the Pontryagin Maximum Principle, we obtain necessary conditions on optimal controls that involve a singular arc. These computations are highlighted thanks to numerical simulations via a direct method. We also study the related optimization problem at steady-state which provides an insight into optimal solutions of the optimal control problem in terms of turnpike phenomenon.

Minimal Time Generation of Density Matrices for a Two-Level Quantum System Driven by Coherent and Incoherent Controls

The article considers a two-level open quantum system whose dynamics is driven by a combination of coherent and incoherent controls. Coherent control enters into the Hamiltonian part of the dynamics whereas incoherent control enters into the dissipative part. The goal is to find controls which move the system from an initial density matrix to a given target density matrix as fast as possible. To achieve this goal, we reformulate the optimal control problem in terms of controlled evolution in the Bloch ball and then apply Pontryagin maximum principle and gradient projection method to numerically find minimal time and optimal coherent and incoherent controls. General method is provided and several examples of initial and target states are explicitly considered.

Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System

In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” by the same authors, general well-posedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous Cahn–Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A, B having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in the quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Frechet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system.

Optimal control in teleoperation systems with time delay: A singular perturbation approach

The main goal of controller design in teleoperation systems is to achieve optimal performance, transparency and stability in presence of factors such as time delay in communication channel and modeling uncertainties. The teleoperation systems usually have complex dynamic. Consequently, differential equation solution of optimal control problem is difficult and complex for them. This paper presents a novel method for designing optimal controller based on singular perturbation framework for these systems. Firstly, we use the Taylor expansion to model the time delay, with considering time delay term; we derive a singular perturbation formulation for the teleoperation system. Using singular perturbation model and Chang decoupling transformation, singularly perturbed differential equations of optimal control problem is decomposed into the reduced order slow and fast differential equations. A formula is obtained that produces the solution of original differential equations of optimal control problem in terms of solutions of the slow and fast reduced order matrix differential equations. The reduced-order differential equations decrease the complexity of the optimal control problem for teleoperation systems. The simulations verify the effectiveness of the proposed control method and excellent performances tracking with high speed and small control signal.

Discrete-Time Fractional Optimal Control

A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of α .

Optimal risk control and dividend distribution policies for a diffusion model with terminal value

In this paper we investigate the optimal risk control and dividend distribution problem for a diffusion model with a terminal value. Usually the insurer cedes risk by means of a reinsurance contract, and pays dividends out dynamically from the surplus. Consider that the insurer is trying to balance risk control and dividend payout in terms of reinsurance and dividend distribution policies. Then the objective is set to make a dynamic choice of reinsurance policy and dividend distribution policy, which maximizes the sum of the expected discounted dividends up to ruin time and the expected discounted terminal value.There are two novelties in this paper. Firstly, we formulate the optimal control problem in terms of general reinsurance control policies. Each of the proportional reinsurance, the excess-of-loss reinsurance and combination of the two can be treated as a special case. It is shown that, under an expected premium principle, the dynamic excess-of-loss reinsurance is of optimal type within the general reinsurance contracts. Secondly, considering the excess-of-loss reinsurance policy and terminal value, we obtain the explicit expressions for the value function and optimal control policies by solving the HJB equation method. At the end of this paper numerical calculations are done to illustrate the influence of the terminal value on the value function and optimal policies as well.

Fast Robust Nanopositioning—A Linear-Matrix-Inequalities-Based Optimal Control Approach

This paper proposes a 2-DOF robust optimal control design method for achieving multiple objectives of resolution, bandwidth, and robustness to modeling uncertainties in nanopositioning systems. The main theoretical contribution of this paper is the formulation of a multiobjective 2-DOF optimal control problem in terms of linear matrix inequalities, which are then solved using standard convex optimization tools. The main distinguishing feature of this approach is the flexibility this method provides in formulating and solving the optimization problems that results in achieving a larger set of performance specifications. It facilitates solving of a certain class of mixed-norm optimization and pole-placement problems that arise naturally in nanopositioning systems. This methodology is demonstrated through experiments on a nanopositioning system that archives performance specifications, which are impossible with 1-DOF designs. Experimental results also demonstrate over 200% improvement in bandwidth of the resulting nanopositioning system over the optimal 1-DOF control designed for the same resolution and robustness specifications.

Optimal control of a symmetric double quantum-dot nanostructure: Analytical results

We study the potential for optimal control of a symmetric double quantum-dot structure interacting with a single pulsed electromagnetic field. We first use the rotating wave and resonant approximations and reduce the dynamics of the system to that of a degenerate three-level $\ensuremath{\Lambda}$-type system. We also formulate the optimal control problem in terms of differential equations that have to be fulfilled by the optimal electromagnetic fields. We then obtain general analytical expressions for the optimal pulse shapes that lead to global maximization of the final population of the target state and of the time-averaged population of the target state in the quantum-dot structure.

From task parameters to motor synergies: A hierarchical framework for approximately-optimal control of redundant manipulators.

We present a hierarchical framework for approximately-optimal control of redundant manipulators. The plant is augmented with a low-level feedback controller, designed to yield input-output behavior that captures the task-relevant aspects of plant dynamics but has reduced dimensionality. This makes it possible to reformulate the optimal control problem in terms of the augmented dynamics, and optimize a high-level feedback controller without running into the curse of dimensionality. The resulting control hierarchy compares favorably to existing methods in robotics. Furthermore we demonstrate a number of similarities to (non-hierarchical) optimal feedback control. Besides its engineering applications, the new framework addresses a key unresolved problem in the neural control of movement. It has long been hypothesized that coordination involves selective control of task parameters via muscle synergies, but the link between these parameters and the synergies capable of controlling them has remained elusive. Our framework provides this missing link.

Nonconvex Duality in Optimal Control

We derive a representation of the minimum cost of an optimal control problem in terms of the upper envelope of generalized semisolutions of the Hamilton--Jacobi equation.

Analytical solution of the optimal laser control problem in two-level systems

The optimal control of two-level systems by time-dependent laser fields is studied using a variational theory. We reformulate the optimal control problem in terms of differential equations, which have to be fulfilled by the optimal fields. We obtain general analytical expressions for the optimal pulse shapes leading to global maximization or minimization of different physical quantities in two-level systems.

Solution of the state-dependent noise optimal control problem in terms of Lyapunov iterations

In this paper we present the solution to the algebraic Riccati-type equation of the state-dependent noise linear–quadratic optimal control problem in terms of algebraic Lyapunov iterations. By properly initializing the sequence of algebraic Lyapunov iterations we got monotonic convergence from above to the positive-definite stabilizing solution of the algebraic Riccati-type equation (the solution that represents the optimal solution to the corresponding optimal control problem). The proposed algorithm requires much less computational efforts than those previously used to solve the same problem.

The solution of a partially observed stochastic optimal control problem in terms of predicted miss

The explicit solution of a partially observed LQ problem driven by a combination of a Wiener process and an unobserved finite-state jump Markov process is given. Applications of the model include guidance problems, where the jump Markov process models evasive maneuvers (acceleration values) of the target, or systems subject to a sequence of failures that can be modeled by a jump Markov process.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>