Time Optimal Control Problem Research Articles

Observability Inequalities on Measurable Sets for the Stokes System and Applications

In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as observability inequalities on space-time measurable sets of positive measure for nonstationary Stokes system. The latter extends the result established recently by Wang and Zhang [SIAM J. Control Optim., 55 (2017), pp. 1862--1886] to the case of observations from subsets of positive measure in both time and space variables. Furthermore, we present their applications in the shape optimization problem, as well as the time optimal control problem for the Stokes system. In particular, we give a positive answer to an open question raised by Privat, Trélat, and Zuazua [Arch. Rational Mech. Anal., 216 (2015), pp. 921--981].

Quasi Time-Optimal Nonlinear Model Predictive Control with Soft Constraints

In many mechatronic applications, controller input costs are negligible and time optimality is of great importance. As a result, the obtained control input has mostly a bang-bang nature, which excite undesired mechanical vibrations, especially in systems with flexible structures. This paper tackles the time-optimal control problem and proposes a novel approach, which explicitly addresses the vibrational behavior in the context of the receding horizon technique. Such technique is a key feature, especially for systems with a time-varying vibrational behavior. In the context of model predictive control (MPC), vibrational behavior is predicted and coped in a soft-constrained formulation, which penalize any violation of undesired vibrations. This formulation enlarges the feasibility on a wide operating range in comparison with a hard-constrained formulation. The closed-loop performance of this approach is demonstrated on a numerical example of stacker crane with high degree of flexibility.

Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions

For a nonlinear finite time optimal control problem, a systematic numerical algorithm to solve the Hamilton-Jacobi equation for a generating function is proposed in this paper. This algorithm allows one to obtain the Taylor series expansion of the generating function up to any prescribed order by solving a sequence of first order ordinary differential equations recursively. Furthermore, the coefficients of the Taylor series expansion of the generating function can be computed exactly under a certain technical condition. Once a generating function is found, it can be used to generate a family of optimal control for different boundary conditions. Since the generating function is computed off-line, the on-demand computational effort for different boundary conditions decreases a lot compared with the conventional method. It is useful to online optimal trajectory generation problems. Numerical examples illustrate the effectiveness of the proposed algorithm.

Time optimal entrainment control for circadian rhythm.

The circadian rhythm functions as a master clock that regulates many physiological processes in humans including sleep, metabolism, hormone secretion, and neurobehavioral processes. Disruption of the circadian rhythm is known to have negative impacts on health. Light is the strongest circadian stimulus that can be used to regulate the circadian phase. In this paper, we consider the mathematical problem of time-optimal circadian (re)entrainment, i.e., computing the lighting schedule to drive a misaligned circadian phase to a reference circadian phase as quickly as possible. We represent the dynamics of the circadian rhythm using the Jewett-Forger-Kronauer (JFK) model, which is a third-order nonlinear differential equation. The time-optimal circadian entrainment problem has been previously solved in settings that involve either a reduced-order JFK model or open-loop optimal solutions. In this paper, we present (1) a general solution for the time-optimal control problem of fastest entrainment that can be applied to the full order JFK model (2) an evaluation of the impacts of model reduction on the solutions of the time-optimal control problem, and (3) optimal feedback control laws for fastest entrainment for the full order Kronauer model and evaluate their robustness under some modeling errors.

Sub-Riemannian Geodesics in $SU(n)/S(U(n-1) \times U(1))$ and Optimal Control of Three Level Quantum Systems

We study the time optimal control problem for the evolution operator of an $n$ -level quantum system. For the considered models, the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. The resulting problem is a sub-Riemannian $K\hbox{--}P$ problem, (as introduced in articles by U. Boscain and by V. Jurdjevic), whose underlying symmetric space is $SU(n)/S(U(n-1) \times U(1))$ . Following a method introduced by F. Albertini and D. D'Alessandro, we consider the action of $S(U(n-1) \times U(1))$ on $SU(n)$ as a conjugation $X \rightarrow KXK^{-1}$ . This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels ( $\Lambda$ -systems). We reduce the latter problem to an integer quadratic optimization problem with linear constraints, which we solve completely for a specific set of final data.

Time-optimal control of heating process subject to phase constraints

The article presents the results of modeling and optimization of temperature fields by the volume of the workpiece during induction heating using a numerical two-dimensional electromagnetic thermal model of the process. Time-optimal control problem of induction heating subject to constraints on maximum permissible temperature is formulated and solved. The obtained optimal control algorithms lead to an increase in the efficiency of induction heaters by reducing total heating time and absence of defects due to technological limitations.

Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.

Application of Pontryagin's minimum principle to Grover's quantum search problem

Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.

Optimal control of epidemic size and duration with limited resources

The total number of infections (epidemic size) and the time needed for the infection to go extinct (epidemic duration) represent two of the main indicators for the severity of infectious disease epidemics in human and livestock. However, few attempts have been made to address the problem of minimizing at the same time the epidemic size and duration from a theoretical point of view by using optimal control theory. Here, we investigate the multi–objective optimal control problem aiming to minimize, through either vaccination or isolation, a suitable combination of epidemic size and duration when both maximum control effort and total amount of resources available during the entire epidemic period are limited. Application of Pontryagin’s Maximum Principle to a Susceptible–Infected–Removed epidemic model, shows that, when the resources are not sufficient to maintain the maximum control effort for the entire duration of the epidemic, the optimal vaccination control admits only bang–bang solutions with one or two switches, while the optimal isolation control admits only bang–bang solutions with one switch. We also find that, especially when the maximum control effort is low, there may exist a trade–off between the minimization of the two objectives. Consideration of this conflict among objectives can be crucial in successfully tackling real–world problems, where different stakeholders with potentially different objectives are involved. Finally, the particular case of the minimum time optimal control problem with limited resources is discussed.

Discrete time optimal control with frequency constraints for non-smooth systems

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) smooth frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of type (b), while less standard in the literature, effectively serve the purpose of describing important properties of inertial actuators and systems. The conjunction of constraints of type (a) and (b) is a relatively new phenomenon in optimal control but is important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability of our theory.

Bang-Bang Property of Time Optimal Control for a Kind of Microwave Heating Problem

In this paper, the bang–bang property and the existence of the solution for the time optimal control of a specific kind of microwave heating problem are studied. The mathematical model for the time optimal control problem (P) of microwave heating is formulated, in which the governing equations of the controlled system are the weak coupling system of Maxwell equations and the heat equation. Based on a new observability estimate of the heat controlled system, the controllability of the system is derived by proving the null controllability of an equivalent system. The existence of the time optimal control of microwave heating is acquired under some stated assumptions, enabling the bang–bang property of the time optimal control problem (P) to be achieved.

Influence of the size of a controllable device on time-optimal rotation generated by a moving internal mass

A two-dimensional time-optimal control problem with a state constraint is studied for a closed mechanical system consisting of a mass point and a solid body that interact via internal forces. It is assumed that the mass point is not allowed to move further away from the body's center of mass than a prescribed distance. A control function is found allowing the body to be turned through a given angle in a minimum time by choosing the velocity of the mass point. In the case of reaching the state constraint, the solution is constructed in explicit form via quadratures representing elliptic integrals. A numerical example of using the derived formulas is given.

Sub-Finsler problem on Cartan group

Left invariant l-infinity sub-Finsler problem on Cartan group is considered as time-optimal control problem. We describe abnormal and singular normal trajectories, then prove that all such trajectories are optimal. We construct the bang-bang flow and obtain upper bounds on the number of switchings on bang-bang and mixed minimizers.

Optimal Control Problems for Certain Linear Fractional-Order Systems Given by Equations with Hilfer Derivative

Two optimal control problems are investigated for linear time-invariant systems of fractional order with lumped parameters, which dynamics is described by equations with Hilfer derivative: control problem with minimal norm and time-optimal control problem with control norm constraint. Controls are considered that are the p-integrable or essentially bounded functions. Investigation is conducted by the method of moments. Correctness and solvability conditions of the problem of moments are obtained for the problem statement considered. The optimal control problems stated are solved analytically for several particular cases and the properties of the solutions are investigated depending on fractional differentiation indices and Hilfer fractional differential operator parameters. The comparison is conducted of the results obtained with the known results for integer-order systems and fractional-order systems described by equations with Riemann–Liouville or Caputo derivative.

The Time-Optimal Control Problem of a Kind of Petrowsky System

In this paper, we consider the time-optimal control problem about a kind of Petrowsky system and its bang-bang property. To solve this problem, we first construct another control problem, whose null controllability is equivalent to the controllability of the time-optimal control problem of the Petrowsky system, and give the necessary condition for the null controllability. Then we show the existence of time-optimal control of the Petrowsky system through minimum sequences, for the null controllability of the constructed control problem is equivalent to the controllability of the time-optimal control of the Petrowsky system. At last, with the null controllability, we obtain the bang-bang property of the time-optimal control of the Petrowsky system by contradiction, moreover, we know the time-optimal control acts on one subset of the boundary of the vibration system.

Modified bang‐bang controller for maximal and minimal time optimal control problems

AbstractRecently, there have been a series of results regarding two time optimal control problems for a class of linear and nonlinear systems ‐ one is to keep the system states within certain bound for the longest time during feedback disruption and the other is to derive the system states to near the origin as fast as possible after feedback recovery, both under bounded control inputs. These are called maximal and minimal time optimal control problems, respectively. In the existing results, a bang‐bang controller has been commonly suggested as the actual implementation of the optimal controller. In this paper, we suggest a modified version of the bang‐bang controller which can also serve as an approximate optimal controller. Our proposed controller provides the (near) optimal performance with (i) possible reduction of a number of switchings; (ii) possible reduction of control input magnitude.

The Bang–Bang Property of Time-Varying Optimal Time Control for Null Controllable Heat Equation

In this paper, we consider bang–bang property for a kind of time-varying time optimal control problem of null controllable heat equation. The study is a continuation of a recent work (Chen et al. in Syst Control Lett 112:18–23, 2018), where an approximate null controllable heat equation was considered. We first establish the equivalence between optimal norm control and optimal time control and then prove the existence of the optimal norm control and of the optimal time control. The time-varying bang–bang property for the optimal time control is finally established.

Semiconcavity results and sensitivity relations for the sub-Riemannian distance

Regularity properties are investigated for the value function of the Bolza optimal control problem with affine dynamic and end-point constraints. In the absence of singular geodesics, we prove the local semiconcavity of the sub-Riemannian distance from a compact set Γ⊂Rn. Such a regularity result was obtained by the second author and L. Rifford in Cannarsa and Rifford (2008) when Γ is a singleton. Furthermore, we derive sensitivity relations for time optimal control problems with general target sets Γ, that is, without imposing any geometric assumptions on Γ.

Variational analysis and optimal control of dynamic unilateral contact models with friction

The target of the present work is to consider the optimal control for a class of evolutionary variational–hemivariational inequalities, which modeling the dynamic viscoelastic unilateral contact problems with normal damped response and friction. After studying the weak solvability of the unilateral contact model, we consider its optimal control by three aspects, the optimal control via external forces and initial conditions, the time optimal control problem and the maximum stay control problem. Finally, the conditions which guarantee the existence of optimal solutions to the corresponding control problems are delivered.

Time-optimal control of multiple unmanned aerial vehicles with human control input

PurposeThe purpose of this paper is to investigate time-optimal control problems for multiple unmanned aerial vehicle (UAV) systems to achieve predefined flying shape.Design/methodology/approachTwo time-optimal protocols are proposed for the situations with or without human control input, respectively. Then, Pontryagin’s minimum principle approach is applied to deal with the time-optimal control problems for UAV systems, where the cost function, the initial and terminal conditions are given in advance. Moreover, necessary conditions are derived to ensure that the given performance index is optimal.FindingsThe effectiveness of the obtained time-optimal control protocols is verified by two contrastive numerical simulation examples. Consequently, the proposed protocols can successfully achieve the prescribed flying shape.Originality/valueThis paper proposes a solution to solve the time-optimal control problems for multiple UAV systems to achieve predefined flying shape.