Bolza Problem Research Articles

Discrete-time optimal control problems with time delay argument: New discrete-time Euler–Lagrange equations with delay

Discrete-time optimal control problems are a crucial type of control problems that deal with a dynamic system evolving in discrete time-steps. This paper introduces a new technique for solving linear discrete-time optimal control problems with state delays, applicable to both finite and infinite time horizons. Our method employs a Riccati matrix equation, optimizing control strategies and ensuring system stability through bounded control inputs. We adopt a Bolza problem for the performance index, which guides the classification of control issues. The technique simplifies problems into manageable Riccati matrix equations using the Euler–Lagrange equations and Pontryagin maximum principle, ensuring stability and necessary condition compliance. The paper validates the approach with numerical examples from quantum mechanics and classical physics, demonstrating its practicality and potential for broader application.

On necessary conditions in the generalized Bolza problem

The paper is devoted to proving necessary optimality conditions for weak/intermediate/strong minimum of a generalized Bolza problem.

Extended Sufficient Conditions of Strong Minimality for the Bolza Problem with Bang Bang Controls. Applications to Space Trajectory Optimization

Extended Sufficient Conditions of Strong Minimality for the Bolza Problem with Bang Bang Controls. Applications to Space Trajectory Optimization

Control and optimization of abstract continuous time evolution inclusions

Abstract Abstract controlled evolution inclusions are revisited in the Banach spaces setting. The existence of solution is established for each selected control. Then, the input–output (or, control-states) multimap is examined and the Lipschitz continuous well posedness is derived. The optimal control of such inclusions handled in terms of a Bolza problem is investigated by means of the so-called P ℱ format of optimization. A strong duality is provided, the existence of an optimal pair is given and the system of optimalty is derived. A Fenchel duality is built and applied to optimal control of convex process of evolution. Finally, it will be shown how the general theory we provided can be applied to a wide class of controled integrodifferental inclusions.

Uniform boundedness for the optimal controls of a discontinuous, non-convex Bolza problem,

We consider a Bolza type optimal control problem of the form [see formula in PDF] Subject to: [see formula in PDF] where Λ(s, y, u) is locally Lipschitz in s, just Borel in (y, u), b has at most a linear growth and both the Lagrangian Λ and the end-point cost function g may take the value +∞. If b ≡ 1, g ≡ 0, (Pt, x) is the classical problem of the Calculus of Variations. We suppose the validity of a slow growth condition in u, introduced by Clarke in 1993, including Lagrangians of the type [see formula in PDF] and [see formula in PDF] and the superlinear case. We show that, if Λ is real valued, any family of optimal pairs (y*, u*) for (Pt,x) whose energy Jt(y*, u*) is equi-boundcd as (t, x) vary in a compact set, has L∞ – equibounded controls. Moreover, if Λ is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on (s, u) ↦ Λ(s, y, u) and requiring a condition on the structure of the effective domain. No convexity, nor local Lipschitzianity is assumed on the variables (y, u). As an application we obtain the local Lipschitz continuity of the value function under slow growth assumptions.

Solutions to the Hamilton-Jacobi equation for state constrained Bolza problems with discontinuous time dependence

This paper provides characterizations of the value function as the unique lower semicontinuous solution, appropriately defined, to the Hamilton-Jacobi equation, for optimal control problems of Bolza type with state constraints. Notable features of our analysis are that it covers problems in which the running cost is possibly non Lipschitz continuous w.r.t. the state variable and the dynamic constraint takes the form of a differential inclusion that is possibly discontinuous w.r.t. time. Alternative characterizations are given, in terms of lower Dini derivative type solutions and of proximal solutions to the Hamilton-Jacobi equation.Distance estimates, which permit us to approximate an arbitrary state trajectory by a state trajectory that strictly satisfies the state constraint, have a key role in the analysis. Earlier research, treating problems in which the running cost is absent or at least Lipschitz continuous w.r.t. the state, makes use of known distance estimates in which ‘approximate’ is interpreted in terms of the L∞ norm. L∞ distance estimates are inadequate, however, for problems in which the running cost may fail to be Lipschitz continuous. We show that, for this broader class of problems, the analysis can still be carried out, if it is based on stronger distance estimates involving the W1,1 norm. The new W1,1 distance estimates, developed here for this purpose, are of independent interest.Motivation for the study of problems with running costs that are not Lipschitz continuous is provided by a ‘growth versus consumption’ problem in neo-classical macro-economics. Our theory provides fresh insights into this much studied problem, by extending the class of initial data points for which a solution can be achieved.

Sufficiency for Weak Minima in Optimal Control Subject to Mixed Constraints

For optimal control problems of Bolza involving time-state-control mixed constraints, containing inequalities and equalities, fixed initial end-point, variable final end-point, and nonlinear dynamics, sufficient conditions for weak minima are derived. The proposed algorithm allows us to avoid hypotheses such as the continuity of the second derivatives of the functions delimiting the problems, the continuity of the optimal controls or the parametrization of the final variable end-point. We also present a relaxation relative to some similar works, in the sense that we arrive essentially to the same conclusions but making weaker assumptions.

Path-Dependent Hamilton--Jacobi Equations with Super-Quadratic Growth in the Gradient and the Vanishing Viscosity Method

The nonexponential Schilder-type theorem in Backhoff-Veraguas, Lacker, and Tangpi [Ann.\Appl. Probab., 30 (2020), pp. 1321--1367] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails a non-Markovian counterpart to the vanishing viscosity method. We show uniqueness of maximal subsolutions for path-dependent viscous Hamilton--Jacobi equations related to convex super-quadratic backward stochastic differential equations. We establish well-posedness for the Hamilton--Jacobi--Bellman equation associated to a Bolza problem of the calculus of variations with path-dependent terminal cost. In particular, uniqueness among lower semicontinuous solutions holds, and state constraints are admitted.

Some Regularity Properties on Bolza problems in the Calculus of Variations

The paper summarizes the main core of the last results that we obtained in [8, 4, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann–Du-Bois Reymond equation.

A Straightforward Sufficiency Proof for a Nonparametric Problem of Bolza in the Calculus of Variations

We study a variable end-points calculus of variations problem of Bolza containing inequality and equality constraints. The proof of the principal theorem of the paper has a direct nature since it is independent of some classical sufficiency approaches invoking the Hamiltonian-Jacobi theory, Riccati equations, fields of extremals or the theory of conjugate points. In contrast, the algorithm employed to prove the principal theorem of the article is based on elementary tools of the real analysis.

Necessary and sufficient conditions of optimality for second order discrete and differential inequalities

Abstract The present paper studies the optimization of the Bolza problem with a system of convex and nonconvex, discrete and differential state variable second-order inequality constraints by deriving necessary and sufficient conditions of optimality. The problem with a system of discrete-approximation inequalities is investigated using the proposed method of discretization and equivalence theorems for subdifferential inclusions, which greatly contributes to the derivation of adjoint discrete inclusions generated by a given system of nonlinear inequality constraints. Furthermore, we formulate sufficient conditions of optimality for the continuous problem by passing to the case of limit. A numerical example is provided to illustrate the theoretical approach’s effectiveness.

Sufficient optimality conditions for the optimal control problem

AbstractThe main purpose of this paper is to establish the sufficient optimality conditions for the optimal controls under some convexity assumptions. For the Bolza problem, under concavity of the Hamiltonian and convexity of the cost functional, the extreme control must be optimal control. However, for the Mayer problem, the cost functional can be relaxed to quasi‐convex and pseudo‐convex. Finally, some examples illustrate these theoretical results and some counter examples show that the convexity assumptions of these results cannot be further weakened in some sense.

Geometric properties of minimizers in the planar three-body problem with two equal masses

It is shown that each lobe of the figure-eight orbit is star-shaped, which indicates that the corresponding polar angle is monotone. In general, it is not clear under which assumptions a trajectory in a minimizer is star-shaped. In this paper, we study minimizers connecting two fixed-ends (i.e. the Bolza problem) in the planar three-body problem with two equal masses. We show that if the Jacobi coordinates of the two fixed-ends are both orthogonal, then for any minimizer, both Jacobi vectors describe a star-shaped curve. If the Jacobi coordinates are orthogonal only on one of the fixed-ends and they are in adjacent closed quadrants, then their polar angles have at most one critical point. As an application, we show the existence and prove some geometric properties of two families of periodic orbits.

Weak Measurable Optimal Controls for the Problems of Bolza

Two sufficiency theorems for parametric and a nonparametric problems of Bolza in optimal control are derived. The dynamics of the problems are nonlinear, the initial and final states are free, and the main results can be applied when nonlinear mixed time-state-control inequality and equality constraints are presented. The deviation between admissible costs and optimal costs around the optimal control is estimated by functionals playing the role of the square of some norms.

Optimal parking management of connected autonomous vehicles: A control-theoretic approach

In this paper we develop a continuous-time stochastic dynamic model for the optimal parking management of connected autonomous vehicles (CAVs) in the presence of multiple parking lots within a given area. Inspired by the well-known Lotka-Volterra equations, a mathematical model is developed to explicitly incorporate the interactions among those parking garages under consideration. Time-dependent parking space availability is considered as the system state, while the dynamic price of parking is naturally used as the control input which can be properly chosen by the parking garage operators from the admissible set. By regulating parking rates, the total demand for parking can be distributed among the set of parking lots under question. Further, we formulate an optimal control problem (called Bolza problem) with the objective of maintaining the availability (managing the demand) of each parking garage at a desired level, which could potentially reduce traffic congestion as well as fuel consumption of CAVs. Based on the necessary conditions of optimality given by Pontryagin’s minimum principle (PMP), we develop a computational algorithm to address the nonlinear optimization problem and formally prove its convergence. A series of Monte Carlo simulations is conducted under various scenarios and the corresponding optimization problems are solved determining the optimal pricing policy for each parking lot. Since the stochastic dynamic model is general and the control inputs, i.e., parking rates, are easy to implement, it is believed that the procedures presented here will shed light on the parking management of CAVs in the near future.

Extended Sufficient Conditions for Strong Minimality in the Bolza Problem: Applications to Space Trajectory Optimization

This paper expands the existing sufficiency results for the strong minimality of an extremal of the Bolza problem. We cover the case where the strict Legendre Clebsh conditions are not strictly verified. An efficient, easy to use, algorithm to prove minimality is provided. It can be used on solutions with bang-bang control and does not require any local controllability property. The interval where sufficiency is provided is maximized over a class of discontinuous Verification Functions determined solving a sequence of Riccati problems. This class of Verification Functions can be adapted to all kind of boundary conditions. The algorithms developed in this paper are applied to space trajectory extremals that exhibit bang-bang control.

Research on attitude control strategy of single engine failure in the first-stage flight phase of new generation launch vehicle

The successful launching of the new generation of launch vehicle is of great practical significance to the development of China Space, and the cryogenic liquid engine commonly used in the launch vehicle has a certain probability of failures, which may affect the success or failure of rocket launching. In this paper, a Bolza problem optimization algorithm based on Radau pseudo-spectral method is proposed to solve the problem of single engine failure in the first-stage flight phase of the new generation launch vehicle by simulating fault injection and taking the attitude angle as optimal control quantity. The simulation results show that this method can effectively eliminate the influence of single engine failure on orbit injection accuracy, achieve the fault absorption of new generation launch vehicle engine, improve the fault tolerance of attitude control system, and further guarantee the system reliability.

Optimal Control of Hybrid Systems via Hybrid Lagrangian Submanifolds

Abstract We develop a geometric interpretation for solutions to the Bolza problem in optimal control for systems subject to hybrid dynamics. In particular, the idea of a hybrid Lagrangian submanifold is introduced. These submanifolds are invariant under the hybrid Hamiltonian flow arising from the extended maximum principle for hybrid systems. Solutions to the hybrid optimal control problem are characterized from backward propagating these submanifolds under the hybrid Hamiltonian flow. This approach side-steps issues of multi-valuedness and singularities. This paper ends with a brief example of the controlled bouncing ball to illustrate the theory.

Controllability and Optimal Control for a Class of Time-Delayed Fractional Stochastic Integro-Differential Systems

In this paper, we study the controllability and optimal control for a class of time-delayed fractional stochastic integro-differential system with Poisson jumps. A set of sufficient conditions is established for complete and approximate controllability by assuming non-Lipschitz conditions and pth mean square norm. We also give an existence of optimal control for Bolza problem. Our result is valid for fractional order $$\alpha >\frac{p-1}{p},\ p\ge 2.$$ Finally, an example is provided to illustrate the efficiency of the obtained theoretical results.

Existence of optimal control for an integro‐differential Bolza problem

SummaryIn the article, we derive an existence theorem for a Bolza problem described by a nonlinear integro‐differential system of Volterra type. We use an approach based on the lower closure theorem for orientor fields due to Cesari and measurable selection theorem of Fillipov type due to Rockafellar. Paper is a continuation of [D. Idczak, S. Walczak, Necessary optimality conditions for an integro‐differential Bolza problem via Dubovitskii‐Miljutin method, DCDS‐B 24 (5) (2019), 2281‐2292; DOI:10.3934/dcdsb.2019095].