Time Optimal Control Problem Research Articles

Одна задача оптимальної швидкодiї для лiнiйної керованої багатозначної системи

УДК 517.9 Розглядається задача оптимальної швидкодії для лінійної керованої багатозначної системи у випадку, коли переріз розв'язку цієї системи збігається з цільовою множиною. Отримано умови розв'язності даної задачі, а також оптимальний час та оптимальні керування. Результати проілюстровано на модельних прикладах.

Advertising in a vaccination campaign: A variable time control problem

The best way to prevent a highly contagious infection such as measles is active immunization with a suitable vaccine. Unfortunately, in the last 10 years in EU there has been a decrease in vaccinations, so that their coverage goes under the minimum level to obtain herd immunity. This is a sanitary emergency and it is connected with the “urban myth” that serious adverse neurological disorders should be attributed to vaccination. Anti-vaccination groups spread this “urban myth” by word-of-mouth communication. We model the evolution of the number of unvaccinated people assuming that a policy-maker can control this dynamics through advertising. The idea of connecting dynamic advertising models and vaccination campaign is recent and it represents some aspects of the issue that have never been explored before. From a mathematical point of view we study a variable final time optimal control problem in which the policy-maker wants to minimize the costs in order to obtain herd immunity in due time. We find the optimal solution and we obtain some prescriptions for the policy-maker.

Asymptotics of the Solution to a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters

The paper continues the authors’ previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball \(\left\{\begin{array}[]{llll}\phantom{\varepsilon^{3}}\dot{x}=y,&x,\,y\in \mathbb{R}^{2},\quad u\in\mathbb{R}^{2},\\ \varepsilon^{3}\dot{y}=Jy+u,&\,\|u\|\leq 1,\quad 0<\varepsilon,\mu\ll 1,\\ x(0)=x_{0}(\varepsilon,\mu)=(x_{0,1},\varepsilon^{3}\mu\xi)^{*},\quad y(0)=y_{ 0},\\ x(T_{\varepsilon,\mu})=0,\quad y(T_{\varepsilon,\mu})=0,\quad T_{\varepsilon, \mu}\to\min,&\end{array}\right.\) where \(J=\left(\begin{array}[]{rr}0&1\\ 0&0\end{array}\right).\) The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix \(J\) at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter \(\mu\). We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence \(\varepsilon^{\gamma}(\varepsilon^{k}+\mu^{k})\), \(0<\gamma<1\).

Two-channel time-optimal control of induction heating process with maximum temperature constraint

The formulation and method of solution of the problem of time-optimal control of induction heating process of an unlimited plate with two control actions on the value of internal heat sources with technological constraint in relation to a one-dimensional model of the temperature field are proposed. The problem is solved under the conditions of a given accuracy of uniform approximation of the final temperature distribution over the thickness of the plate to the required. The method of finite integral transformations is used to search for the input-output characteristics of an object with distributed parameters with two control actions. The preliminary parameterization of control actions based on analytical optimality conditions in the form of the Pontryagin maximum principle is used. At the next stage reduction is performed to the problem of semi-infinite optimization, the solution of which is found using the alternance method. The alternance properties of the final resulting temperature state at the end of the optimal process lead to a basic system of relations, which, if there is additional information about the shape of the temperature distribution curve, is reduced to a system of equations that can be solved. An example of solving the problem of time-optimal control of temperature field of an unlimited plate with two offices is carried out in two stages. At first stage the case of induction heating without maximum temperature constraints is considered, at the second stage is carried out on the basis of the results of the first stage to obtain the solution subject to the limitation on the maximum temperature of the heated billet.

Time optimal control of two-level quantum systems

The objective of time-optimal control that helps to minimize relaxation losses, is the evolution of a quantum state from a given initial mixed state to a final target mixed state in minimum time. In this paper, we study a time-optimal control problem of the dynamic of a pure two-level system with unbounded control using Pontryagin's minimum principle and obtain the minimal time for some initial and final states. The results will apply to basically all qubit systems that one can consider such as NMR spectroscopy, trapped ions, superconducting qubits, etc. We also show that these results hold for pure states, and only the direction nˆ is important in the evolution of a quantum state. In this work, the problem of computing minimum time to produce any unitary transformation Uf∈SU(2) is reduced to finding the minimum time to steer the system from an initial to a final state.

Time-optimal bang-bang control for the driven spin-boson system

In this paper we investigate the application of optimal control of a single qubit coupled to an ohmic heat bath. For the weak bath coupling regime, we derive a Bloch-Redfield master equation describing the evolution of the qubit state parametrized by vectors in the Bloch sphere. By use of the optimal control methodology we determine the field that generates an arbitrary single qubit rotation in minimum time. In particular we develop an efficient method for solving the time-optimal control problem, namely the implementation of $X$ gate, which consists of a rotation about the $\stackrel{P\vec}{ox}$ axis through angle $\ensuremath{\pi}$. The time-optimal control problem requires a bounded control. If upper and lower bounds are imposed for the external control, the optimal solution is of bang-bang type and switches from the upper to the lower values of the control bounds. We compare our results using the techniques of automatic differentiation to compute the gradient for the cost functional with some known results using Pontryagin's minimum principle. We use an alternative numerical approach for the optimization strategy.

Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature

We consider a time-optimal control problem on the plane with a circular vectogram of velocities and a non-convex target set with a boundary having a finite number of points of discontinuity of curvature. We study the problem of identifying and constructing scattering curves that form a singular set of the optimal result function in the case when the points of discontinuity of curvature have one-sided curvatures of different signs. It is shown that these points belong to pseudo-vertices that are characteristic points of the boundary of the target set, which are responsible for the generation of branches of a singular set. The structure of scattering curves and the optimal trajectories starting from them, which fall in the neighborhood of the pseudo-vertex, is investigated. A characteristic feature of the case under study is revealed, consisting in the fact that one pseudo-vertex can generate two different branches of a singular set. The equation of the tangent to the smoothness points of the scattering curve is derived. A scheme is proposed for constructing a singular set, based on the construction of integral curves for first-order differential equations in normal form, the right-hand sides of which are determined by the geometry of the boundary of the target in neighborhoods of the pseudo-vertices. The results obtained are illustrated by the example of solving the control problem when the target set is a one-dimensional manifold.

Time-optimal control of quantum lambda systems in the KP configuration

We consider the problem of time-optimal control for a three-level quantum system where one level is coupled by the control field to the lowest two, which are not coupled to each other. A bound is assumed on the norm of the control at every time. Such a problem belongs to the class of KP sub-Riemannian problems for which we can perform a symmetry reduction and reduce to a Riemannian problem on the associated quotient space. We prove several properties of such a quotient space in our case, including the fact that it is an example of an almost-Kähler manifold that is not Kähler. We provide the explicit form of the optimal controls for any unitary transformation on the lowest two levels and discuss the geometric and practical implications of this result.

An Indirect Method for Regular State-Constrained Optimal Control Problems in Flow Fields

This article concerns an indirect method to solve state-constrained optimal control problems. The dynamics of the controlled system is given by ordinary differential equations encompassing the effect of a steady flow field in which the moving object is immersed. The proposed method is based on the maximum principle in Gamkrelidze's form. At the core of the approach is a regularity hypothesis which entails the continuity of the measure Lagrange multiplier associated with the state constraint. This property plays key role in shaping a properly modified shooting method to solve the two-point boundary value problem resulting from the maximum principle. Illustrative applications to time optimal control problems are considered and results of numerical experiments are provided.

Optimal Sizing and Control of Battery Energy Storage Systems for Hybrid Turboelectric Aircraft

&lt;div class="section abstract"&gt;&lt;div class="htmlview paragraph"&gt;Hybrid-electric gas turbine generators are considered a promising technology for more efficient and sustainable air transportation. The Ohio State University is leading the NASA University Leadership Initiative (ULI) &lt;i&gt;Electric Propulsion: Challenges and Opportunities&lt;/i&gt;, focused on the design and demonstration of advanced components and systems to enable high-efficiency hybrid turboelectric powertrains in regional aircraft to be deployed in 2030. Within this large effort, the team is optimizing the design of the battery energy storage system (ESS) and, concurrently, developing a supervisory energy management strategy for the hybrid system to reduce fuel burn while mitigating the impact on the ESS life. In this paper, an energy-based model was developed to predict the performance of a battery-hybrid turboelectric distributed-propulsion (BHTeDP) regional jet. A study was conducted to elucidate the effects of ESS sizing and cell selection on the optimal power split between the turbogenerators (TGs) and ESS. To this extent, the supervisory energy management strategy is formulated into a discrete time optimal control problem and solved via dynamic programming. The performance of BHTeDP was compared to a turboelectric distributed-propulsion (TeDP) next-gen aircraft that assumes improvements in weight, drag, and engine efficiency consistent with regional jet entering operation in 2035.&lt;/div&gt;&lt;/div&gt;

Regular path-constrained time-optimal control problems in three-dimensional flow fields

This article concerns a class of time-optimal state constrained control problems with dynamics defined by an ordinary differential equation involving a three-dimensional steady flow vector field. The problem is solved by virtue of an indirect method based on the maximum principle in Gamkrelidze’s form. The proposed computational method essentially uses a certain regularity condition imposed on the data of the problem. A regularity assumption guarantees the continuity of the measure multiplier associated with the state constraint and ensures the appropriate behavior of the corresponding numerical procedure which, in general, consists of computing the entire field of extremals for the problem in question. Several examples of vector fields are considered to illustrate the computational approach.

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.

A simple proof of the discrete time geometric Pontryagin maximum principle on smooth manifolds

We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in Chang (2011) in the context of continuous-time optimal control.

Time-optimality by distance-optimality for parabolic control systems

The equivalence of time-optimal and distance-optimal control problems is shown for a class of parabolic control systems. Based on this equivalence, an approach for the efficient algorithmic solution of time-optimal control problems is investigated. Numerical examples are provided to illustrate that the approach works well in practice.

Optimal Tyre Management of a Formula One car

Optimal control calculations are used to study the effect of tyre wear on race car performance. This is achieved by solving a minimum lap time optimal control problem over multiple laps using a dynamical model of a Formula One car. A previously developed thermodynamic model is enhanced by adding an additional state for the carcass temperature of the tyres. The tyre grip is modelled as a function of the tyres’ wear and temperature. Grip reduces when tyres get worn, or the tyres are not operated within their optimal temperature window. Overheating the tyres can accelerate wear, which in turn, degrades performance. The optimal control problem solver needs to ‘manage’ the state of tyres throughout a race (not just a single lap) to ensure that optimal race performance is achieved. At some point during a race a pit stop may be required to change worn tyres so that tyre grip can be restored. It is essential to understand the wear characteristics of various tyre compounds in order to determine the point when the time needed for a pit stop is justified in terms of subsequent racing performance.

On the construction of nearly time optimal continuous feedback laws around switching manifolds

In this paper, we address the question of the construction of a nearly time optimal feedback law for a minimum time optimal control problem, which is robust with respect to internal and external perturbations. For this purpose we take as starting point an optimal synthesis, which is a suitable collection of optimal trajectories. The construction we exhibit depends exclusively on the initial data obtained from the optimal feedback which is assumed to be known.

APPROXIMATION OF FRACTIONAL RESOLVENTS AND APPLICATIONS TO TIME OPTIMAL CONTROL PROBLEMS

We investigate the approximation of fractional resolvents, extending and improving some corresponding results on semigroups and resolvents. As applications, we utilize the approach of Meyer approximation to analyze the time optimal control problem of a Riemann-Liouville fractional system without Lipschitz continuity. A fractional diffusion model is also presented to confirm our theoretical findings.

Minimal time sliding mode control for evolution equations in Hilbert spaces

This work is concerned with the time optimal control problem for evolution equations in Hilbert spaces. The attention is focused on the maximum principle for the time optimal controllers having the dimension smaller that of the state system, in particular for minimal time sliding mode controllers, which is one of the novelties of this paper. We provide the characterization of the controllers by the optimality conditions determined for some general cases. The proofs rely on a set of hypotheses meant to cover a large class of applications. Examples of control problems governed by parabolic equations with potential and drift terms, porous media equation or reaction-diffusion systems with linear and nonlinear perturbations, describing real world processes, are presented at the end.

Local Controllability and a Family of Maximum Principles for a Free Time Optimal Control Problem

We introduce the concept of local controllability for a control system of ordinary differential equations with free time and provide sufficient conditions for its controllability. As a direct consequence, for time-optimal problems, we obtain necessary conditions for a local infimum---this notion generalizes that of an optimal trajectory. These conditions constitute a family of relations, each of which has the form of a maximum principle. Examples are given to demonstrate the meaningfulness of the necessary conditions obtained in the present paper, which generalize and strengthen the Pontryagin maximum principle.

Fréchet Analysis and Sensitivity Relations for the Optimal Time Problem

In this paper, we first present formulas for computing the Frechet subdifferentials and Frechet singular subdifferentials of the minimal time function for a differential inclusion in R n with a general target K. These formulas are characterized in terms of Frechet normal cones to a sub-level set of and a level set of the minimized Hamiltonian associated with the differential inclusion. Our results extend, improve and complement to existing results. An application to sensitivity analysis for a class of time optimal control problems is also given.