Showing papers in "Siam Journal on Control and Optimization in 2014"
TL;DR: This work describes a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems.
Abstract: We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.
145 citations
TL;DR: These definitions capture precisely the requirements on effective supervisory control in networked discrete event systems in the sense that a networked supervisor exists if and only if both network controllability and network observability are satisfied.
Abstract: In this paper, we investigate supervisory control of networked discrete event systems. Such systems are now widely used in practice. The key feature of a networked discrete event system is communication delays and losses in observation and control. Without addressing communication delays and losses in networked systems, no control will be effective. We propose to use some special observation mappings to deal with observation delays and losses. Based on these mappings, we define network observability. We also consider control delays and losses in networked discrete event systems and define network controllability. We show that these definitions capture precisely the requirements on effective supervisory control in networked discrete event systems in the sense that a networked supervisor exists if and only if both network controllability and network observability are satisfied.
135 citations
TL;DR: The value process is characterized as the unique viscosity solution of the corresponding path dependent Bellman-Isaacs equation, a notion recently introduced by Ekren et al.
Abstract: In this paper we study a two person zero sum stochastic differential game in weak formulation. Unlike the standard literature, which uses strategy type controls, the weak formulation allows us to consider the game with control against control. We shall prove the existence of game value under natural conditions. Another main feature of the paper is that we allow for non-Markovian structure, and thus the game value is a random process. We characterize the value process as the unique viscosity solution of the corresponding path dependent Bellman-Isaacs equation, a notion recently introduced by Ekren et al. (Ann. Probab., 42 (2014), pp. 204-236) and Ekren, Touzi, and Zhang (Stochastic Process., to appear; preprint, arXiv:1210.0006v2; preprint, arXiv:1210.0007v2).
125 citations
TL;DR: A simple payoff-based learning rule that is completely decentralized and that leads to an efficient configuration of actions in any $n$-person finite strategic-form game with generic payoffs that follows the theme of exploration versus exploitation and is hence stochastic in nature.
Abstract: We propose a simple payoff-based learning rule that is completely decentralized and that leads to an efficient configuration of actions in any $n$-person finite strategic-form game with generic payoffs. The algorithm follows the theme of exploration versus exploitation and is hence stochastic in nature. We prove that if all agents adhere to this algorithm, then the agents will select the action profile that maximizes the sum of the agents' payoffs a high percentage of time. The algorithm requires no communication. Agents respond solely to changes in their own realized payoffs, which are affected by the actions of other agents in the system in ways that they do not necessarily understand. The method can be applied to the optimization of complex systems with many distributed components, such as the routing of information in networks and the design and control of wind farms. The proof of the proposed learning algorithm relies on the theory of large deviations for perturbed Markov chains.
119 citations
Journal Article•
TL;DR: In this article, a new framework for approximately solving flow problems in capacitated, undirected graphs and applying it to provide asymptotically faster algorithms for the maximum s-t flow and maximum concurrent multicommodity flow problems was introduced.
Abstract: In this paper, we introduce a new framework for approximately solving flow problems in capacitated, undirected graphs and apply it to provide asymptotically faster algorithms for the maximum s-t flow and maximum concurrent multicommodity flow problems. For graphs with n vertices and m edges, it allows us to find an e-approximate maximum s-t flow in time O(m1+o(1)e-2), improving on the previous best bound of O(mn1/3poly(e-1)). Applying the same framework in the multicommodity setting solves a maximum concurrent multicommodity flow problem with k commodities in O(m1+o(1)e-2k2) time, improving on the existing bound of O(m4/3poly(k>,e-1)).Our algorithms utilize several new technical tools that we believe may be of independent interest:• We give a non-Euclidean generalization of gradient descent and provide bounds on its performance. Using this, we show how to reduce approximate maximum flow and maximum concurrent flow to oblivious routing.• We define and provide an efficient construction of a new type of flow sparsifier. Previous sparsifier constructions approximately preserved the size of cuts and, by duality, the value of the maximum flows as well. However, they did not provide any direct way to route flows in the sparsifier G' back in the original graph G, leading to a longstanding gap between the efficacy of sparsification on flow and cut problems. We ameliorate this by constructing a sparsifier G' that can be embedded (very efficiently) into G with low congestion, allowing one to transfer flows from G' back to G.• We give the first almost-linear-time construction of an O(mo(1))-competitive oblivious routing scheme. No previous such algorithm ran in time better than Ω(mn). By reducing the running time to almost-linear, our work provides a powerful new primitive for constructing very fast graph algorithms.The interested reader is referred to the full version of the paper [8] for a more complete treatment of these results.
117 citations
TL;DR: Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration of the class of strictly dissipative systems under consideration.
Abstract: We investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints. Considering a class of strictly dissipative systems, we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property. Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration.
116 citations
TL;DR: This work solves the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information and derives relations between the inner and the outer accuracy of the primal and dual problems.
Abstract: We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution of the augmented Lagrangian primal problem is hard to compute in practice, we solve this problem up to some given inner accuracy. We derive relations between the inner and the outer accuracy of the primal and dual problems and we give a full convergence rate analysis for both gradient and fast gradient algorithms. We provide estimates on the primal and dual suboptimality and on primal feasibility violation of the generated approximate primal and dual solutions. Our analysis relies on the Lipschitz property of the dual function and on inexact dual gradients. We also discuss implementation aspects of the proposed algor...
115 citations
TL;DR: A modular approach is followed, where subobservers are connected in cascade to achieve a desired exponential convergence rate (chain observer) and relationships among the error decay rate, the bound on the measurement delays, the observer gains, and the Lipschitz constants of the system are presented.
Abstract: This paper presents a method for designing state observers with exponential error decay for nonlinear systems whose output measurements are affected by known time-varying delays. A modular approach is followed, where subobservers are connected in cascade to achieve a desired exponential convergence rate (chain observer). When the delay is small, a single-step observer is sufficient to carry out the goal. Two or more subobservers are needed in the the presence of large delays. The observer employs delay-dependent time-varying gains to achieve the desired exponential error decay. The proposed approach allows to deal with vector output measurements, where each output component can be affected by a different delay. Relationships among the error decay rate, the bound on the measurement delays, the observer gains, and the Lipschitz constants of the system are presented. The method is illustrated on the synchronization problem of continuous-time hyperchaotic systems with buffered measurements.
96 citations
TL;DR: A fundamental error equivalence drastically simplifies the a posteriori error analysis for optimal control problems, and it basically remains to apply error estimators for the linear state and adjoint problem.
Abstract: We derive a unifying framework for the a posteriori error analysis of control constrained linear-quadratic optimal control problems. We consider finite element discretizations with discretized and nondiscretized control. A fundamental error equivalence drastically simplifies the a posteriori error analysis for optimal control problems. It basically remains to apply error estimators for the linear state and adjoint problem. We give several examples, including stabilized discretizations, and investigate the quality of the estimators and the performance of the adaptive iteration by selected numerical experiments.
93 citations
TL;DR: In this paper, the theory of robust output regulation of distributed parameter systems with infinite-dimensional exosystems is extended for plants with unbounded control and observation, and a new way of defining an internal model is introduced.
Abstract: In this paper the theory of robust output regulation of distributed parameter systems with infinite-dimensional exosystems is extended for plants with unbounded control and observation. As the main result, we present the internal model principle for linear infinite-dimensional systems with unbounded input and output operators. We do this for two different definitions of an internal model found in the literature, namely, the $p$-copy internal model and the $\mathcal{G}$-conditions. We also introduce a new way of defining an internal model for infinite-dimensional systems. The theoretic results are illustrated with an example where we consider robust output tracking for a one-dimensional heat equation with boundary control and pointwise measurements.
90 citations
TL;DR: In this paper, the authors considered a linear quadratic stochastic two-person zero-sum differential game, where both players are allowed to appear in both drift and diffusion of the state equation, and the weighting matrices in the performance functional are not assumed to be definite/non-singular.
Abstract: In this paper, we consider a linear quadratic stochastic two-person zero-sum differential game. The controls for both players are allowed to appear in both drift and diffusion of the state equation. The weighting matrices in the performance functional are not assumed to be definite/non-singular. The existence of an open-loop saddle point is characterized by the existence of an adapted solution to a linear forward-backward stochastic differential equation with constraints, together with a convexity-concavity condition, and the existence of a closed-loop saddle point is characterized by the existence of a regular solution to a Riccati differential equation. It turns out that there is a significant difference between open-loop and closed-loop saddle points. Also, it is found that there is an essential feature that prevents a linear quadratic optimal control problem from being a special case of linear quadratic two-person zero-sum differential games.
TL;DR: In this paper, the optimal boundary control of a time-discrete Cahn--Hilliard--Navier--Stokes system is studied and a general class of free energy potentials is considered which, in particular, includes the double-obstacle potential.
Abstract: In this paper, the optimal boundary control of a time-discrete Cahn--Hilliard--Navier--Stokes system is studied. A general class of free energy potentials is considered which, in particular, includes the double-obstacle potential. The latter homogeneous free energy density yields an optimal control problem for a family of coupled systems, which result from a time discretization of a variational inequality of fourth order and the Navier--Stokes equation. The existence of an optimal solution to the time-discrete control problem as well as an approximate version is established. The latter approximation is obtained by mollifying the Moreau--Yosida approximation of the double-obstacle potential. First order optimality conditions for the mollified problems are given, and in addition to the convergence of optimal controls of the mollified problems to an optimal control of the original problem, first order optimality conditions for the original problem are derived through a limit process. The newly derived statio...
TL;DR: In this paper, the authors introduce the notion of path-complete graph Lyapunov functions for approximation of the joint spectral radius of directed graphs and derive asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratics, and maximum/minimum-of-quadratics LQF functions.
Abstract: We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratic, and maximum/minimum-of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We derive approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs. This provides worst-case performance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.
TL;DR: The boundary controllability of a class of one-dimensional degenerate equations is studied and sharp observability estimates for these equations are proved using nonharmonic Fourier series.
Abstract: The boundary controllability of a class of one-dimensional degenerate equations is studied in this paper. The novelty of this work is that the control acts through the part of the boundary where degeneracy occurs. First we consider a class of degenerate hyperbolic equations. Then, we prove sharp observability estimates for these equations using nonharmonic Fourier series. The transmutation method yields a result for the corresponding class of degenerate parabolic equations.
TL;DR: A novel time-varying (nonadaptive) design scheme is successfully proposed to finite-time stabilizing controllers and it is shown that if the design parameters and design functions in the designed controller are suitably chosen, then all the signals of the closed-loop system are globally stabilized.
Abstract: This paper focuses on the global finite-time stabilization via time-varying state-feedback for a class of uncertain time-varying nonlinear systems. The problem is rather difficult to solve, mainly due to the presence of more serious unknowns and time-variations and higher nonlinearities in the systems under investigation, particularly those in the control coefficients of the systems. To solve the problem, by skillfully combining time-varying methods and finite-time control theory in the existing literature, a novel time-varying (nonadaptive) design scheme is successfully proposed to finite-time stabilizing controllers. The remarkable feature of the design scheme is the selection of time-varying design functions which can capture the unknowns and time-variations in the systems as time increases. It is shown that if the design parameters and design functions in the designed controller are suitably chosen, then all the signals (i.e., the system states and control input) of the closed-loop system are globally...
TL;DR: A directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems and it is used to establish structural properties of the minimizer.
Abstract: A directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems. It allows for controls which are localized in space, where the spatial support is independent of time. Well-posedness of the optimal control problems is established and the optimality system is derived. It is used to establish structural properties of the minimizer. An a priori error analysis for finite element discretization is obtained, and numerical results illustrate the effects of the proposed cost functional and the convergence results.
TL;DR: In this paper, the authors consider stochastic differential games with $N$ players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time average cost with quadratic running cost.
Abstract: We consider stochastic differential games with $N$ players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls ...
TL;DR: The Saint-Venant system for the regulation of water flows in a network of canals is investigated, with finite-time boundary stabilization shown to occur roughly in a twice longer time when only one boundary feedback law is available.
Abstract: We investigate the finite-time boundary stabilization of a one-dimensional first order quasilinear hyperbolic system of diagonal form on [0,1]. The dynamics of both boundary controls are governed by a finite-time stable ODE. The solutions of the closed-loop system issuing from small initial data in Lip$([0,1])$ are shown to exist for all times and to reach the null equilibrium state in finite time. When only one boundary feedback law is available, a finite-time stabilization is shown to occur roughly in a twice longer time. The above feedback strategy is then applied to the Saint-Venant system for the regulation of water flows in a network of canals.
TL;DR: The main results are to identify the right Hamilton--Jacobi--Bellman equation and to provide the maximal and minimal solutions, as well as conditions for uniqueness.
Abstract: This article is a continuation of a previous work where we studied infinite horizon control problems for which the dynamic, running cost, and control space may be different in two half-spaces of some Euclidian space $\mathbb{R}^N$. In this article we extend our results in several directions: (i) to more general domains; (ii) to consideration of finite horizon control problems; (iii) to weakening the controllability assumptions. We use a Bellman approach and our main results are to identify the right Hamilton--Jacobi--Bellman equation (and, in particular, the right conditions to be put on the interfaces separating the regions where the dynamic and running cost are different) and to provide the maximal and minimal solutions, as well as conditions for uniqueness. We also provide stability results for such equations.
TL;DR: It is proved that the partial differential equation model is well-posed and an exponentially stable closed-loop system can be obtained for a set of system parameters of zero Lebesgue measure.
Abstract: Models for piezoelectric beams and structures with piezoelectric patches generally ignore magnetic effects. This is because the magnetic energy has a relatively small effect on the overall dynamics. Piezoelectric beam models are known to be exactly observable and can be exponentially stabilized in the energy space by using a mechanical feedback controller. In this paper, a variational approach is used to derive a model for a piezoelectric beam that includes magnetic effects. It is proved that the partial differential equation model is well-posed. Magnetic effects have a strong effect on the stabilizability of the control system. For almost all system parameters the piezoelectric beam can be strongly stabilized, but it is not exponentially stabilizable in the energy space. Strong stabilization is achieved using only electrical feedback. Furthermore, using the same electrical feedback, an exponentially stable closed-loop system can be obtained for a set of system parameters of zero Lebesgue measure. These r...
TL;DR: In this paper, the number of critical formations of point agents on a line is estimated using Morse theory and complex algebraic geometry, and it is shown that there are at least $2N-1$ equilibrium points and at most $3^{N −1}$ isolated equilibria.
Abstract: Formation shape control for a collection of point agents is concerned with devising decentralized control laws which ensure that the formation will move so that certain interagent distances approximate prescribed values as closely as possible. Such laws are often derived using steepest descent of a potential function which is invariant under translation and rotation, and then critical formations are those that are fixed under the evolution of the decentralized control dynamics, i.e., those corresponding to equilibrium points of the control dynamics. Using a specific and frequently used potential function for formation control, this paper introduces tools from Morse theory and complex algebraic geometry to estimate the number of critical formations of $N$ agents on a line. We show that there are at least $2N-1$ equilibrium points and at most $3^{N-1}$ isolated equilibria. Moreover, bounds on the number of equilibrium points with a $k$-dimensional stable manifold (the so-called Morse-index) are established....
TL;DR: In this paper, a probabilistic solution of a control problem with a state constraint by means of a backward stochastic differential equation (BSDE) was proposed, which possesses a singular terminal condition.
Abstract: We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a backward stochastic differential equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [Stochastic Process. Appl., 116 (2006), pp. 2014--2056] and [Ann. Probab., 35 (2007), pp. 1071--1117]. We perform a verification and discuss special cases for which the control problem has explicit solutions.
TL;DR: The fact that discontinuities are overcome by the sampling and holding process enlarges greatly the possibility of finding successful controllers for retarded nonlinear systems, by means of control Lyapunov--Krasovskii functionals.
Abstract: In this paper, the problem of the stabilization in the sample-and-hold sense for fully nonlinear systems with an arbitrary number of arbitrary discrete as well as of distributed time delays is studied. It is shown that steepest descent feedbacks, continuous or not, induced by Lyapunov--Krasovskii functionals in a suitable (large) class, are stabilizers in the sample-and-hold sense. The fact that discontinuities are overcome by the sampling and holding process enlarges greatly the possibility of finding successful controllers for retarded nonlinear systems, by means of control Lyapunov--Krasovskii functionals.
TL;DR: It is shown that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case.
Abstract: There are several results on the stability of nonlinear positive systems in the presence of time delays. However, most of them assume that the delays are constant. This paper considers time-varying, possibly unbounded, delays and establishes asymptotic stability and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, we present a necessary and sufficient condition for delay-independent stability of continuous-time positive systems whose vector fields are cooperative and homogeneous. We show that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays. For various classes of time delays, we are able to derive explicit expressions that quantify the decay rates of positive systems. We also provide the corresponding counterparts for discrete-time positive systems whose vector fields are nondecreasing and homogeneous.
TL;DR: It is demonstrated the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDE-constrained control problem.
Abstract: A control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied. The control is the right-hand side forcing function and the objective of control is a standard matching functional. A recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system. When sufficient conditions on certain kernel functions and the volume constraints hold, the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived. We demonstrate the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDE-constrained control problem. We also define continuous and discontinuous Galerkin finite element discretizations of the optimality system for which we derive a priori error estimates. Numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained f...
TL;DR: New control functions are derived for linear difference equations with delay, which can be used to construct a nontrivial solution of a boundary value problem with zero boundary conditions.
Abstract: New control functions are derived for linear difference equations with delay, which can be used to construct a nontrivial solution of a boundary value problem with zero boundary conditions. Later, a special case of invariant linear subspace is considered and corresponding control functions are constructed. Results for weakly nonlinear problems are also discussed.
TL;DR: The impact of various random effects on the underlying systems for almost sure and $p$th-moment stability is revealed and insight is provided on stability and stabilization of switching jump diffusion systems.
Abstract: This work focuses on regime-switching jump diffusions, which include three classes of random processes, Brownian motions, Poisson processes, and Markov chains. First, a scalar linear system is treated as a benchmark model. Then stabilization of systems with one-sided linear growth is considered. Next, nonlinear systems that have a finite explosion time are treated, in which regularization (explosion suppression) and stabilization are achieved by introducing appropriate diffusions together with Poisson and Markov chain perturbations. This work reveals the impact of various random effects on the underlying systems for almost sure and $p$th-moment stability and provides insight on stability and stabilization of switching jump diffusion systems.
TL;DR: In this paper, the authors studied the stochastic optimal control problem of fully coupled forward-backward stochastically differential equations (FBSDEs) and proved that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions.
Abstract: In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton--Jacobi--Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85--138. We emphasize that when $\sigma$ depends on the second component of the solution $(Y, Z)$ of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence...
TL;DR: A Luenberger-like observer is proposed which is shown to converge asymptotically to the actual state, provided the initial value of the state estimation error is small enough.
Abstract: This paper deals with the stability and observer design for Lur'e systems with multivalued nonlinearities, which are not necessarily monotone or time-invariant. Such differential inclusions model the motion of state trajectories which are constrained to evolve inside time-varying nonconvex sets. Using Lyapunov-based analysis, sufficient conditions are proposed for local stability in such systems, while specifying the basin of attraction. If the sets governing the motion of state trajectories are moving with bounded variation, then the resulting state trajectories are also of bounded variation, and unlike the convex case, the stability conditions depend on the size of jumps allowed in the sets. Based on the stability analysis, a Luenberger-like observer is proposed which is shown to converge asymptotically to the actual state, provided the initial value of the state estimation error is small enough. In addition, a practically convergent state estimator, based on the high-gain approach, is designed to reduc...
TL;DR: Stability of the solutions of optimal control problems in measure spaces governed by semilinear elliptic equations is analyzed and highly nonlinear terms can be incorporated by utilizing an $L^\infty (Omega) regularity result for solutions of the first order necessary optimality conditions.
Abstract: Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient second order optimality conditions are obtained as well. On the basis of the sufficient conditions, stability of the solutions is analyzed. Highly nonlinear terms can be incorporated by utilizing an $L^\infty (\Omega)$ regularity result for solutions of the first order necessary optimality conditions.