In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of variation. In the paper we prove that absolutely continuous functions that are solutions of Mayer problem satisfy the Lipschitz condition. Using this fact we show that the value function is a bilateral solution of Hamilton–Jacobi equation. Moreover, we prove that continuity or the local Lipschitz condition of the function of final cost g is inherited by the value function. Our results allow to state the theorem about existence and uniqueness of bilateral solutions in the class of functions that are bounded from below and satisfy the local Lipschitz condition. In proving the main results we use recently derived necessary optimality conditions of Loewen–Rockafellar [P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 32 (1994) 442–470; P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 35 (1997) 2050–2069].

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1)\{½}. It follows that, under this assumption, our starting question has a positive answer.

We consider the variational problem

        inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1},

for α, β ∈ [0, 1], α + β ≤ 1, where λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.

We study properties of the functional

\begin{eqnarray} \mathscr{F}_{{\rm loc}}(u,\Omega):=\inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(\nabla u_{j})\udx\, \left| \!\!\begin{array}{rl} & (u_{j})\subset W_{{\rmloc}}^{1,r}\left(\Omega, \RN\right) \\ & u_{j}\tostar u\,\,\textrm{in}\BV\left(\Omega, \RN\right) \end{array} \right. \bigg\}, \end{eqnarray}$\begin{array}{ccc}{F}_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{:}\mathrm{=}\underset{\mathrm{\left(}{\mathit{u}}_{\mathit{j}}\mathrm{\right)}}{\inf }\left\{\right.\underset{\mathit{j}\mathrm{\to }\mathrm{\infty }}{\lim \inf }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }{\mathit{u}}_{\mathit{j}}\mathrm{\right)}\mathrm{d}\mathit{x}\left|\begin{array}{c}\\ & \\ & \end{array}\right.\left\}\right.\mathit{,}& & \end{array}$

\hbox{$r\in[1,\frac{n}{n-1})$}$\mathit{r}\mathrm{\in }\mathrm{\left[}\mathrm{1}\mathit{,}\frac{\mathit{n}}{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

\begin{equation*} \scF_{{\rm loc}}(u,\Omega) \geq \int_{\Omega}f(\nabla u (x))\ud x + \int_{\Omega}\finf\bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|, \end{equation*}$F_{\mathrm{loc}}\mathrm{\left(}\mathit{u,\Omega }\mathrm{\right)}\mathrm{\ge }{\mathrm{\int }}_{\mathit{\Omega }}\mathit{f}\mathrm{\left(}\mathrm{\nabla }\mathit{u}\mathrm{\right(}\mathit{x}\mathrm{\left)}\mathrm{\right)}\mathrm{d}\mathit{x}\mathrm{+}{\mathrm{\int }}_{\mathit{\Omega }}{\mathit{f}}^{\mathrm{\infty }}\left(\right.\frac{{\mathit{D}}^{\mathit{s}}\mathit{u}}{\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}}\left)\right.\mathrm{\left|}{\mathit{D}}^{\mathit{s}}\mathit{u}\mathrm{\right|}\mathit{,}$

\hbox{$ f^{\infty}(\xi):= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$}${\mathit{f}}^{\mathrm{\infty }}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{:}\mathrm{=}{\overline{\lim }}_{\mathit{t}\mathrm{\to }\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{t\xi }\mathrm{\right)}\mathit{/}\mathit{t}$

We introduce the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external policy maker, and we propagate its effect for the number N of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This will be realized by considering cost functionals including L1-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of Γ-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u,E), Hölder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set is obtained under suitable closeness assumptions on the eigenvalues of the bulk energies.

We prove uniform continuity ofradially symmetric vector minimizersuA(x) = UA(|x|)to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on aball BR ⊂ ℝd,among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using ajointly convex lsc L∗∗ : ℝm×ℝ → [0,∞]withL∗∗(S,·) evenand superlinear. Besides such basic hypotheses,L∗∗(·,·) is assumed to satisfy alsoa geometrical constraint, which we callquasi − scalar; the simplest example being thebiradial caseL∗∗(|u(x)|,|Du(x)|).Complete liberty is given forL∗∗(S,λ)to take the ∞ value, so that our minimization problem implicitly also representse.g. distributed-parameteroptimal control problems, onconstrained domains, under PDEs or inclusions inexplicit or implicit form. While generic radial functionsu(x) = U(|x|) inthis Sobolev space oscillate wildly as |x| → 0, our minimizingprofile-curve UA(·) is, incontrast, absolutely continuous andtame, in the sense that its“static level” L∗∗(UA(r),0)always increases with r, a original feature of our result.

In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error in L2(Ω) of the control variable is h | log h | in dimensions 2 and 3.

This paper investigates the output controllability problem of temporal Boolean networkswith inputs (control nodes) and outputs (controlled nodes). A temporal Boolean network isa logical dynamic system describing cellular networks with time delays. Using semi-tensorproduct of matrices, the temporal Boolean networks can be converted into discrete timelinear dynamic systems. Some necessary and sufficient conditions on the outputcontrollability via two kinds of inputs are obtained by providingcorresponding reachable sets. Two examples are given to illustrate the obtainedresults.

The adaptive stabilization is investigated for a class of coupled PDE-ODE systems with multiple uncertainties. The presence of the multiple uncertainties and the interaction between the sub-systems makes the systems to be considered more general and representative, and moreover it may result in the ineffectiveness of the conventional methods on this topic. Motivated by the existing literature, an infinite-dimensional backsteppping transformation with new kernel functions is first introduced to change the original system into a target system, from which the control design and performance analysis of the original system will become quite convenient. Then, by certainty equivalence principle and Lyapunov method, an adaptive stabilizing controller is successfully constructed, which guarantees that all the closed-loop system states are bounded while the original system states converging to zero. A simulation example is provided to validate the proposed method.

This paper is addressed to proving a new Carleman estimate for stochastic parabolic equations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X. Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm. 5.2), one extra gradient term involving in that estimate is eliminated. Also, our improved Carleman estimate is established by virtue of the known Carleman estimate for deterministic parabolic equations. As its application, we prove the existence of insensitizing controls for backward stochastic parabolic equations. As usual, this insensitizing control problem can be reduced to a partial controllability problem for a suitable cascade system governed by a backward and a forward stochastic parabolic equation. In order to solve the latter controllability problem, we need to use our improved Carleman estimate to establish a suitable observability inequality for some linear cascade stochastic parabolic system, while the known Carleman estimate for forward stochastic parabolic equations seems not enough to derive the desired inequality.

In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of the calculus of variations are described. Necessary optimality conditions in the main problem of the calculus of variations and in the problem of Bolza in the nonsmooth case are derived. Examples comparing presented results with other approaches to nonsmooth problems of the calculus of variations are given.

In this note we prove compactness for the Cahn–Hilliard functional without assumingcoercivity of the multi-well potential.

In this paper, we study the boundary penalty method for optimal control of unsteady Navier–Stokes type system that has been proposed as an alternative for Dirichlet boundary control. Existence and uniqueness of solutions are demonstrated and existence of optimal control for a class of optimal control problems is established. The asymptotic behavior of solution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergence of solutions of penalized control problem to the corresponding solutions of the Dirichlet control problem, as the penalty parameter goes to zero. We also derive an optimality system and determine optimal solutions. In order to illustrate the theoretical results and the practical utility of control, we numerically address the problem of controlling unsteady convection with Soret effect using a gradient-based method. Numerical results show the effectiveness of the approach.

We consider a damped abstract second order evolution equation with an additionalvanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, wedo not assume the operator defining the main damping to be bounded. First, using aconstructive frequency domain method coupled with a decomposition of frequencies and theintroduction of a new variable, we show that if the limit system is exponentially stable,then this evolutionary system is uniformly − with respect to the calibration parameter −exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decayestimates of the underlying semigroup provided such decay estimates hold for the limitsystem. Finally, we discuss some applications of our results; in particular, the case ofboundary damping mechanisms is accounted for, which was not possible in the earlier workmentioned above.

This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier−Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-α equations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α → 0+ to a null control of the Navier−Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.

We show that we can reconstruct two coefficients of a wave equation by a single boundary measurement of the solution. The identification and reconstruction are based on Krein’s inverse spectral theory for the first coefficient and on the Gelfand−Levitan theory for the second. To do so we use spectral estimation to extract the first spectrum and then interpolation to map the second one. The control of the solution is also studied.

The Euler−Poinsot rigid bodymotion is a standard mechanical system and it is a model for left-invariant Riemannianmetrics on SO(3). In this article using theSerret−Andoyer variables weparameterize the solutions and compute the Jacobi fields in relation with the conjugatelocus evaluation. Moreover, the metric can be restricted to a 2D-surface, and theconjugate points of this metric are evaluated using recent works on surfaces ofrevolution. Another related 2D-metric on S2 associated to the dynamics of spin particles withIsing coupling is analysed using both geometric techniques and numerical simulations.

In this paper we consider a new kind of Mumford–Shah functionalE(u, Ω) for mapsu : ℝm → ℝnwith m ≥ n. The most important novelty is that theenergy features a singular set Su ofcodimension greater than one, defined through the theory of distributional jacobians.After recalling the basic definitions and some well established results, we prove anapproximation property for the energy E(u, Ω) viaΓ −convergence, in the same spirit of the work by Ambrosio andTortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math.43 (1990) 999–1036].

A dual-weighted residual approach for goal-oriented adaptive finite elements for a class of optimal control problems for elliptic variational inequalities is studied. The development is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Also, a priori bounds for C-stationary points and associated multipliers are derived. Details on the numerical realization of the adaptive concept are provided and a report on numerical tests including the critical cases of biactivity are presented.