Bolza Optimal Control Problem Research Articles

Existence of Optimal Control for Nonlinear Fokker–Planck Equations in \(\boldsymbol{L^1(\mathbb{R}^d)}\).

This work is concerned with the existence of optimal controllers for the Bolza optimal control problem governed by the nonlinear Fokker–Planck equation in with control input in the drift term. The solution to the control state system is a weak (mild) solution obtained from a vanishing viscosity approximation scheme. One obtains in particular the existence for the stochastic optimal control problem governed by McKean–Vlasov SDEs. For this problem, one proves the existence of a stochastic Markov optimal controller in feedback form.

Minimax and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Time-Delay Systems

The paper deals with a Bolza optimal control problem for a dynamical system which motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional in this problem, the Cauchy problem for the Hamilton-Jacobi-Bellman equation with coinvariant derivatives is considered. Minimax and viscosity solutions of this problem are studied. It is proved that both of these solutions exist, are unique and coincide with the value functional.

Regularity and necessary conditions for a Bolza optimal control problem

We consider a Bolza optimal control problem whose Lagrangian, possibly extended valued, may be discontinuous in the state and control variable so that optimal solutions are not supposed to necessarily satisfy the Maximum Principle. Given an optimal trajectory-control pair, we prove that it satisfies a new Erdmann – Du Bois-Reymond type condition, and show that, from this condition, it is possible to derive boundedness properties of the optimal control and a Lipschitz regularity result for the optimal state arc, just imposing general growth assumptions (allowing some almost linear growth behaviors).

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.

Stability of Solutions to Hamilton–Jacobi Equations Under State Constraints

In the present paper, we investigate stability of solutions of Hamilton---Jacobi---Bellman equations under state constraints by studying stability of value functions of a suitable family of Bolza optimal control problems under state constraints. The stability is guaranteed by the classical assumptions imposed on Hamiltonians and an inward-pointing condition on state constraints.

Variational Approach to Second-Order Sufficient Optimality Conditions in Optimal Control

We consider a Bolza optimal control problem with convex pure state constraints and general control constraints and propose new second-order sufficient optimality conditions for a weak local minimum that can be applied in a general framework. In particular, the reference control may be merely measurable and the dynamics may be discontinuous in the time variable. Our approach is based on a new decomposition result for control constraints involving second-order tangents.

Local regularity of the value function in optimal control

It is well-known that the value function V of a Bolza optimal control problem fails to be everywhere differentiable. In this paper, however, we show that, if V is proximally subdifferentiable at (t,x), then it is smooth on a neighborhood of (t,x). Our result yields that V stays smooth on a neighborhood of any optimal trajectory starting at a point where the proximal subdifferential is nonempty. This leads to sufficient conditions for the regularity of optimal trajectories and optimal controls.

Optimal control for evolution equations with memory

In this paper, we investigate the existence and regularity of solutions for Bolza optimal control problems in infinite dimension governed by a class of semilinear evolution equations. Our results apply to systems exhibiting hereditary properties, as heat propagation in real conductors and isothermal viscoelasticity, described by equations with memory terms which account for the past history of the variables in play.

Wavelet Collocation Method for Optimal Control Problems

A Haar wavelet technique is discussed as a method for discretizing the nonlinear system equations for optimal control problems. The technique is used to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. A nonlinear programming solver can then be used to solve optimal control problems that are rather general in form. Here, general Bolza optimal control problems with state and control constraints are considered. Examples of two kinds of optimal control problems, continuous and discrete, are solved. The results are compared to those obtained by using other collocation methods.

On Bolza optimal control problems with constraints

We provide sufficient conditions for the existence and Lipschitz continuity of solutions to the constrained Bolza optimal control problem <p align="center"> $\text{minimize}\quad \int_0^T L(x(t),u(t))\dt + l(x(T))$ <p align="left" class="times"> over all trajectory / control pairs $(x,u)$, subject to the state equation <p align="center"> x'(t)=$f(x(t),u(t)) $ for a.e. $t\in [0,T]$<br> $u(t)\in U $ for a.e. $t\in [0,T]$<br> $x(t)\in K $ for every $t\in [0,T]$<br> $x(0)\in Q_0\.$ <p align="left" class="times"> The main feature of our problem is the unboundedness of $f(x,U)$ and the absence of superlinear growth conditions for $L$. Such classical assumptions are here replaced by conditions on the Hamiltonian that can be satisfied, for instance, by some Lagrangians with no growth. This paper extends our previous results in <i> Existence and Lipschitz regularity of solutions to Bolza problems in optimal control</i> to the state constrained case.

Hölder continuity of minimizers for the state constrained Boza problem

AbstractRegularity of minimizers for the Bolza optimal control problem has been extensively studied by several authors (cf. For instance [2–5] and the references contained therein). Here we give a generalization of some results of [5] by using an approach introduced in [3]. We consider a Bolza problem under state constraints and show that, if the Hamiltonian function is smooth enough and enjoys some monotonicity properties, then both the adjoint state and optimal trajectory enjoy Hölder type regularity. We also provide sufficient conditions for Hölder regularity of optimal controls. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.

Lipschitzianity of optimal trajectories for the Bolza optimal control problem

In this paper we investigate Lipschitz continuity of optimal solutions for the Bolza optimal control problem under Tonelli’s type growth condition. Such regularity being a consequence of normal necessary conditions for optimality, we propose new sufficient conditions for normality of state-constrained nonsmooth maximum principles for absolutely continuous optimal trajectories. Furthermore we show that for unconstrained problems any minimizing sequence of controls can be slightly modified to get a new minimizing sequence with nice boundedness properties. Finally, we provide a sufficient condition for Lipschitzianity of optimal trajectories for Bolza optimal control problems with end point constraints and extend a result from (J. Math. Anal. Appl. 143, 301–316, 1989) on Lipschitzianity of minimizers for a classical problem of the calculus of variations with discontinuous Lagrangian to the nonautonomous case.

Second-Order Necessary Conditions and Sufficient Conditions Applied to Continuous-Thrust Trajectories

Introduction I N optimal control theory, requiring the first variation of the performance functional to vanish leads to well-known first-order necessary conditions (NC) for an optimal solution.1 These NC allow one to identify candidates for optimality, called stationary or extremal solutions, to distinguish them from solutions that have been proven to be optimal. To determine whether a stationary solution is indeed optimal, one must also test the second-order Jacobi no-conjugate-point NC, which applies if the trajectory is smooth. Also, one can formulate sufficient conditions (SC) that, if satisfied, guarantee that the solution is at least locally optimal. In this Note, a procedure developed by Jo and Prussing2−4 for testing second-order NC and SC is streamlined and applied to an example optimal continuous-thrust trajectory with multiple terminal constraints that yields a different type of result compared to previous examples in Refs. 2, 4, and 5. The procedure is based on earlier work by Wood5,6 that derives new, less restrictive SC for a weak local minimum of the Bolza optimal control problem. However, those SC require that the solution of a matrix Riccati equation be bounded. This is difficult to test numerically because a bounded but rapidly increasing solution can stop the numerical integration and give the false impression that the solution is unbounded. The procedure described and illustrated in this Note replaces the test for an unbounded matrix5,6 by a test for a (scalar) determinant being zero.

On singularities of value function for Bolza optimal control problem

We study the set of points of nondifferentiability, called the singular set, of the value function of a Bolza optimal control problem. The value function is a viscosity solution to an associated Hamilton–Jacobi equation. The method of characteristics associates to this equation a Hamiltonian system, that in turn can be used to study the propagation of singularities of the value function. In particular, we obtain an extension of the Rankine–Hugoniot type condition, which is well-known in the conservation law theory.

Direct Trajectory Optimization by a Chebyshev Pseudospectral Method

We present a Chebyshev pseudospectral method for directly solving a generic Bolza optimal control problem with state and control constraints. This method employs Nth-degree Lagrange polynomial approximations for the stateand control variables with the values of these variables at the Chebyshev-Gauss-Lobatto (CGL) points as the expansion coefficients. This process yields a nonlinear programming problem (NLP) with the state and control values at the CGL points as unknown NLP parameters. Numerical examples demonstrate that this method yields more accurate results than those obtained from the traditional collocation methods.

Semicontinuous Solutions of Hamilton–Jacobi–Bellman Equations with Degenerate State Constraints

In this paper, the value function of a Bolza optimal control problem with state constraints is characterized as the unique lower semicontinuous solution of a Hamilton–Jacobi equation. The state constraints are given by an arbitrary closed set with possibly empty interior. In particular, Soner's inward pointing condition is extended here to the case of degenerate state constraints.

Procedure for Applying Second-Order Conditions in Optimal Control Problems

A recent advance in sufe cient conditions for a weak local minimum in the Bolza optimal control problem is used to develop a practical procedure for applying second-order necessary conditions and sufe cient conditions. For a system with n state variables, a transition matrix method is used to transform a test for the unboundedness of an n £ n matrix solution of a Riccati equation into a test for a scalar being zero. This allows routine testing of second-order conditions, including the Jacobi no-conjugate-point necessary condition. Four example problems are analyzed: a simple minimum-time problem, the shortest path between two points on a sphere, a multiobjective spacecraft trajectory optimization, and an application of Hamilton’ s Principle to a circular orbit in an inversesquare gravitational e eld. In those examples for which second-order conditions are violated and an analytical solution does not exist, a genetic algorithm is used to determine a near-optimal solution.

$\Bbb L^2$ Sufficient Conditions for End-Constrained Optimal Control Problems With Inputs in a Polyhedron

An $\hbox{{\bbb L}}^2$-local optimality sufficiency theorem is proved for a class of structured infinite-dimensional nonconvex programs with constraints of the form $u\in\Omega$ and h(u)=0, where $\Omega$ is a set of Lebesgue measurable essentially bounded vector-valued functions $u(\cdot ): [0,1]\rightarrow \hbox{{\bbb R}}^m$ with range in a polyhedron U, and h is a smooth map of the space of essentially bounded functions $u(\cdot )$ into ${\bbb R}^k$. The sufficiency theorem is based on formal counterparts of the finite-dimensional Karush--Kuhn--Tucker sufficient conditions in a Cartesian product of polyhedra, a strengthened variant of Pontryagin's necessary condition, and structure and continuity conditions on the first and second differentials of the objective function and equality constraint functions. The new sufficient conditions are directly applicable to nonconvex continuous-time Bolza optimal control problems with control-quadratic Hamiltonians, unqualified affine inequality constraints on vector-valued control inputs, and equality constraints on the terminal state vector or equivalent isoperimetric constraints on integrals of functions depending on the state and control variables.

Optimal Control of a Ship for a Course-Changing Maneuver

The steering control of a ship during a course-changing maneuver is formulated as a Bolza optimal control problem, which is solved via the sequential gradient-restoration algorithm (SGRA). Nonlinear differential equations describing the yaw dynamics of a steering ship are employed as the differential constraints, and both amplitude and slew rate limits on the rudder are imposed. Two performance indices are minimized: one measures the time integral of the squared course deviation between the actual ship course and a target course; the other measures the time integral of the absolute course deviation. Numerical results indicate that a smooth transition from the initial set course to the target course is achievable, with a trade-off between the speed of response and the amount of course angle overshoot.