Family Of Optimal Control Problems Research Articles

Variance Optimization and Control Regularity for Mean-Field Dynamics

Abstract We study a family of optimal control problems in which one aims at minimizing a cost that mixes a quadratic control penalization and the variance of the system, both for finitely many agents and for the mean-field dynamics as their number goes to infinity. While solutions of the discrete problem always exist in a unique and explicit form, the behavior of their macroscopic counterparts is very sensitive to the magnitude of the time horizon and penalization parameter. When one minimizes the final variance, there always exists a Lipschitz-in-space optimal controls for the infinite dimensional problem, which can be obtained as a suitable extension of the optimal controls for the finite-dimensional problems. The same holds true for variance maximizations whenever the time horizon is sufficiently small. On the contrary, for large final times (or equivalently for small penalizations of the control cost), it can be proven that there does not exist Lipschitz-regular optimal controls for the macroscopic problem.

Exact controllability of linear mean‐field stochastic systems and observability inequality for mean‐field backward stochastic differential equations

AbstractThis paper is concerned with the exact controllability of linear mean‐field stochastic systems with time‐variant random coefficients. We prove that the exact controllability, the validity of the observability inequality for the dual equation, the unique solvability of a family of optimal control problems, the unique solvability of a family of mean‐field forward‐backward stochastic differential equations (MF‐FBSDEs), and the unique solvability of a family of norm optimal control problems are all equivalent. Therefore, some approaches are provided to investigate the issue of exact controllability.

A Level-Set Approach for Stochastic Optimal Control Problems Under Controlled-Loss Constraints

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton–Jacobi–Bellman equation usually calls for strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in the absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then solve the latter by a level-set approach. With this approach, state constraints are managed through an exact penalization technique.

Uniform Convergence of the Solutions of Riccati Equations for a Family of Optimal Control Problems

The uniform convergence of the solutions of a parameterized family of Riccati differential equations which arise in the context of optimal control problems for systems described by a family of linear time-varying evolution equations is considered on the infinite time interval in Hilbert space. Some sufficient conditions and an illustrative example are given for uniform convergence of the solutions of a family of the Riccati differential equations. The uniform convergence of the solutions of the Riccati differential equations with respect to parameters is important for applications to problems in adaptive control of stochastic evolution systems.

Homogenization of a transmission problem with Hamilton–Jacobi equations and a two-scale interface. Effective transmission conditions

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of ε while the period is of the order of ε2. As ε→0, the interfaces tend to a straight line Γ. We study the asymptotic behavior of the value function as ε→0. We prove that the value function tends to the solution of Hamilton–Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations.

Solving quantum optimal control problems using Clebsch variables and Lin constraints

Clebsch variables (and Lin constraints) are applied to the study of a class of optimal control problems for affine-controlled quantum systems. The optimal control problem will be modelled with controls defined on an auxiliary space where the dynamical group of the system acts freely. The reciprocity between both theories: the classical theory defined by the objective functional and the quantum system, is established by using a suitable version of Lagrange’s multipliers theorem and a geometrical interpretation of the constraints of the system as defining a subspace of horizontal curves in an associated bundle. It is shown how the solutions of the variational problem defined by the objective functional determine solutions of the quantum problem. Then a new way of obtaining explicit solutions for a family of optimal control problems for affine-controlled quantum systems (finite or infinite dimensional) is obtained. One of its main advantages, is the the use of Clebsch variables allows to compute such solutions from solutions of invariant problems that can often be computed explicitly. This procedure can be presented as an algorithm that can be applied to a large class of systems.Finally, some simple examples, spin control, a simple quantum Hamiltonian with an ‘Elroy beanie’ type classical model and a controlled one-dimensional quantum harmonic oscillator, illustrating the main features of the theory, will be discussed.

Effective transmission conditions for Hamilton–Jacobi equations defined on two domains separated by an oscillatory interface

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Γε. The oscillations of the interface have small period and amplitude, both of the order of ε, and the interfaces Γε tend to a straight line Γ. We study the asymptotic behavior as ε→0. We prove that the value function tends to the solution of Hamilton–Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations of Γε.

Optimal control of two coupled spinning particles in the Euler–Lagrange picture

A family of optimal control problems for a single and two coupled spinning particles in the Euler–Lagrange formalism is discussed. A characteristic of such problems is that the equations controlling the system are implicit and a reduction procedure to deal with them must be carried out. The reduction of the implicit control equations arising in these problems will be discussed in the slightly more general setting of implicit equations defined by invariant one-forms on Lie groups. As an example the first order differential equations describing the extremal solutions of an optimal control problem for a single spinning particle, obtained by using Pontryagin’s Maximum Principle (PMP), will be found and shown to be completely integrable. Then, again using PMP, solutions for the problem of two coupled spinning particles will be characterized as solutions of a system of coupled non-linear matrix differential equations. The reduction of the implicit system will show that the reduced space for them is the product of the space of states for the independent systems, implying the absence of ‘entanglement’ in this instance. Finally, it will be shown that, in the case of identical systems, the degree three matrix polynomial differential equations determined by the optimal feedback law, constitute a completely integrable Hamiltonian system and some of its solutions are described explicitly.

Convergence of optimal control problems governed by second kind parabolic variational inequalities

We consider a family of optimal control problems where the control variable is given by a boundary condition of Neumann type. This family is governed by parabolic variational inequalities of the second kind. We prove the strong convergence of the optimal controls and state systems associated to this family to a similar optimal control problem. This work solves the open problem left by the authors in IFIP TC7 CSMO2011.

Optimal Control of Vehicular Formations With Nearest Neighbor Interactions

We consider the design of optimal localized feedback gains for one-dimensional formations in which vehicles only use information from their immediate neighbors. The control objective is to enhance coherence of the formation by making it behave like a rigid lattice. For the single-integrator model with symmetric gains, we establish convexity, implying that the globally optimal controller can be computed efficiently. We also identify a class of convex problems for double-integrators by restricting the controller to symmetric position and uniform diagonal velocity gains. To obtain the optimal non-symmetric gains for both the single- and the double-integrator models, we solve a parameterized family of optimal control problems ranging from an easily solvable problem to the problem of interest as the underlying parameter increases. When this parameter is kept small, we employ perturbation analysis to decouple the matrix equations that result from the optimality conditions, thereby rendering the unique optimal feedback gain. This solution is used to initialize a homotopy-based Newton's method to find the optimal localized gain. To investigate the performance of localized controllers, we examine how the coherence of large-scale stochastically forced formations scales with the number of vehicles. We establish several explicit scaling relationships and show that the best performance is achieved by a localized controller that is both non-symmetric and spatially-varying.

Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind

Let u g be the unique solution of a parabolic variational inequality of second kind, with a given g . Using a regularization method, we prove, for all g 1 and g 2 , a monotony property between μ u g 1 + ( 1 − μ ) u g 2 and u μ g 1 + ( 1 − μ ) g 2 for μ ∈ [ 0 , 1 ] . This allowed us to prove the existence and uniqueness results to a family of optimal control problems over g for each heat transfer coefficient h > 0 , associated with the Newton law, and of another optimal control problem associated with a Dirichlet boundary condition. We prove also, when h → + ∞ , the strong convergence of the optimal controls and states associated with this family of optimal control problems with the Newton law to that of the optimal control problem associated with a Dirichlet boundary condition.

Virtual control regularization of state constrained linear quadratic optimal control problems

A numerical method for linear quadratic optimal control problems with pure state constraints is analyzed. Using the virtual control concept introduced by Cherednichenko et al. (Inverse Probl. 24:1---21, 2008) and Krumbiegel and Rosch (Control Cybern. 37(2):369---392, 2008), the state constrained optimal control problem is embedded into a family of optimal control problems with mixed control-state constraints using a regularization parameter ?>0. It is shown that the solutions of the problems with mixed control-state constraints converge to the solution of the state constrained problem in the L 2 norm as ? tends to zero. The regularized problems can be solved by a semi-smooth Newton method for every ?>0 and thus the solution of the original state constrained problem can be approximated arbitrarily close as ? approaches zero. Two numerical examples with benchmark problems are provided.

On Dynamic Programming in Economic Models Governed by DDEs

A family of optimal control problems for economic models, where state variables are driven by delay differential equations (DDEs) and subject to constraints, is treated by Bellman's dynamic programming in infinite dimensional spaces. An existence theorem is provided for the associated Hamilton-Jacobi-Bellman (HJB) equation: the value function of the control problem solves the HJB equation in a suitable sense (although such value function cannot be computed explicitly). An AK model with vintage capital and an advertising model with delay effect are taken as examples.

Performance Assessment of Air-Core Linear Permanent-Magnet Synchronous Motors

Performance assessment is crucial during the motor design process as a basis for comparison between different candidate designs. The objective of this paper is to determine how quickly an air-core linear permanent-magnet synchronous motor can perform a point-to-point positioning task while not exceeding a given limit on allowable average power dissipation. The magnetic field in the air gap is obtained by solving the Poisson equation, and an explicit representation for the Lorentz force is derived. Using this force model, performance limits are identified by solving a family of optimal control problems numerically.

Homogenization of Dirichlet optimal control problems with exact partial controllability constraints

We consider an elliptic distributed quadratic optimal control problem with exact controllability constraints on a part of the domain which, in turn, is parametrized by a small parameter e. The quadratic tracking type functional is defined on the remaining part of the domain. We thus consider a family of optimal control problems with state equality constraints. The purpose of this paper is to study the asymptotic limit of the optimal control problems as the parameter e tends to zero. The analysis presented is in the spirit of the direct approach of the calculus of variations. This is achieved in the framework of relaxed problems. We finally apply the procedure to an optimal control problem on a perforated domain with holes of critical size. It is shown that a strange term in the terminology of Cioranescu and Murat (Prog. Nonlinear Diff. Eq. Appl., Vol. 31, Birkhauser-Verlag, Boston, 1997, pp. 49-93) appears in the limiting homogenized problem.

Phase Portrait Bifurcation in One-Parameter Family of Optimal Control Problems

A two-dimensional optimal control problem is considered with a performance index depending upon a scalar parameter. For the zero value of the parameter one has the well known Fuller problem with the chattering phenomenon involving infinitely many switching points in finite time (Fuller, 1963). A maximum principle based preliminary analysis shows that such a regime takes place up to a certain critical value of the parameter, after which one has a two-switching regime involving a first order singular arc. The optimality of the above mentioned regimes is proved using dynamic programming. A group-invariant analysis of the Bellman equation, similar to that in (Wonham, 1963) reveals the structure of the smooth Bellman function involving several unknown constants which can be found numerically from an algebraic system of equations. From the same system one gets the coefficients of the switching, curve which consists of two generally nonsymmetric half-parabolas, as in (Marchall, 1975).

Mode reconstruction in a one-parameter family of optimal control problems

A two-dimensional free-time optimal control problem with an integral performance index which depends on a scalar parameter is considered. For zero value of the parameter the problem reduces to the well-known Fuller problem with the chattering phenomenon involving infinitely many switching points in a finite time [1]. A preliminary analysis, based on the maximum principle, shows that such a regime is maintained up to a certain critical value of the parameter, after which one has a two-switching regime involving a first-order singular arc. The optimality of the above-mentioned regimes is proved using dynamic programming. A group-invariant analysis of Bellman's equation, similar to that in [2], reveals the structure of the twice differentiable Bellman's function involving several unknown constants. These are found numerically from an algebraic system of equations.

On a certain family of optimal control problems with phase constraint

On a certain family of optimal control problems with phase constraint

An optimization model of the Stefan problem

It was shown in [1, 2, 4], when approximating the parabolic Stefan problem by a family of optimalcontrol problems in which the controlling parameter is the form of the region in which the state-temperature function of the liquid phase is defined, that the classical solution of the Stefan problem is the limit of the solutions of the corresponding approximation problems in the metrics of approximate functional spaces. The structure of the optimal-control approximation problems which model the Stefan problem is investigated, the necessary conditions for these problems to be solvable are obtained, and a proof of the convergence of the proposed approximate methods of solving them is given.

ε-optimal control of random parabolic differential equations by an elliptic approximation

We consider an optimal control problem for an abstract ITO equation on a Gelfand triple of Hilbert spaces. This control problem is approximated by means of a family of optimal control problems for elliptic systems