On the existence of optimal controls

A control process described by Eq. (1) is considered. Existence theorems for controls which minimize a functional of a general type are given by using weak compactness criteria.The authors are indebted to Professors R. Conti and L. D. Berkovitz for their helpful comments.     

On the existence of optimal controls

Using a recent result due to Berkovitz, we prove the existence of an optimal control in a broad class of problems, under relatively mild conditions.     

On existence of optimal controls

An optimal control is shown to exist for a system when the Hamiltonian is a strictly convex function of the control. It is proven that a system satisfying this condition must have state equations that are linear in the control and a cost functional whose integrand is strictly convex in the control.The authors wish to thank Professor G. Leitmann for fruitful discussion of this problem.     

On the approximation of optimal stochastic controls

The stochastic maximum principle gives a necessary condition for the optimal control problem for diffusions. If the controlled diffusion is approximated by a controlled Markov chain, and if approximating controls are chosen to maximize a Hamiltonian for the chain, then it is shown using weak convergence that the chains converge to a diffusion with a control satisfying the necessary condition of the maximum principle, and the corresponding costs also converge.     

On the regularity of solutions of optimal transportation problems

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.     

On the existence of optimal partially observed controls

A stochastic control problem whose dynamics are only partially observed is solved. In earlier literature it was conjectured that for such problems an optimal relaxed control exists. In this article we prove that for the problem under consideration the optimal relaxed control exists and is the weak limit of a minimizing sequence of ordinary controls. Making use of the special discrete nature of the observations and of the special form of the drift function the existence of an optimal ordinary control is derived.The general partially observed control problem is then approximated by a sequence of problems of the above form, i.e., with discrete observations. In this way the existence of an ordinary optimal control is derived for the general problem.During part of his work on this topic the author was a guest of the SFB 72 of the Deutsche Forschungsgemeinschaft of the University of Bonn.The author's work was partially supported by the Deutsche Forschungsgemeinschaft within the SFB 72 of the University of Bonn.     

On the boundedness of optimal controls in infinite-horizon problems

A class of infinite-horizon optimal control problems that arise in economic applications is considered. A theorem on the nonemptiness and boundedness of the set of optimal controls is proved by the method of finite-horizon approximations and the apparatus of the Pontryagin maximum principle. As an example, a simple model of optimal economic growth with a renewable resource is considered.     

On the existence of optimal controls for nonlinear infinite dimensional systems

We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.     

Interior regularity of optimal transport paths

In a previous paper [12], we considered problems for which the cost of transporting one probability measure to another is given by a transport path rather than a transport map. In this model overlapping transport is frequently more economical. In the present article we study the interior regularity properties of such optimal transport paths. We prove that an optimal transport path of finite cost is rectifiable and simply a finite union of line segments near each interior point of the path.Received: 30 October 2002, Accepted: 26 August 2003, Published online: 15 October 2003Mathematics Subject Classification (2000): 90B06, 49Q20.     

On the existence of optimal controls for quasilinear parabolic partial differential equations

A class of systems governed by quasilinear parabolic partial differential equations with first boundary conditions is considered. Existence of solutions for this class of systems and theira priori estimates are established. Further, a theorem on the existence of optimal controls for the corresponding control problem is obtained. Its proof is based on Filippov's implicit functions lemma. The control restraint setU is taken as a measurable multifunction.The authors wish to thank Professor L. Cesari for his most valuable comments and suggestions. In fact, a condition assumed in the original version of this paper was substantially relaxed by him. For details, see Remark 4.1.     

Partial regularity for optimal transport maps

We prove that, for general cost functions on R ⁿ , or for the cost d ²/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.     

On optimal regularity of free boundary problems and a conjecture of De Giorgi

We present a general theory to study optimal regularity for a large class of nonlinear elliptic systems satisfying general boundary conditions and in the presence of a geometric transmission condition on the free boundary. As an application we give a full positive answer to a conjecture of De Giorgi on the analyticity of local minimizers of the Mumford‐Shah functional. © 2004 Wiley Periodicals, Inc.     

On the theory of singular optimal controls in dynamic systems with control delay

An optimal control problem with a control delay is considered, and a more broad class of singular (in classical sense) controls is investigated. Various sequences of necessary conditions for the optimality of singular controls in recurrent form are obtained. These optimality conditions include analogues of the Kelley, Kopp–Moyer, R. Gabasov, and equality-type conditions. In the proof of the main results, the variation of the control is defined using Legendre polynomials.     

Fractal regularity results on optimal irrigation patterns

In this paper the problem of the regularity, i.e. fractal behaviour, of the minima of the branched transport problem is addressed. We show that, under suitable conditions on the irrigated measure, the minima present a fractal regularity, that is on a given branch of length l the number of branches bifurcating from it whose length is comparable with ε   can be estimated both from above and below by l/ε$l/\epsilon$.     

On the existence of optimal coefficient controls for a nonlinear Neumann boundary value problem

We study the solvability of an optimal control problem for a nonlinear elliptic equation with the Neumann conditions on the boundary for the case in which the coefficients in the main part of the differential operator play the role of control functions. We show that this problem is solvable in the class of generalized-solenoidal matrices.