A deterministic affine-quadratic optimal control problem is considered. Due to the nature of the problem, optimal controls exist under some very mild conditions. Further, it is shown that under some assumptions, the optimal control is unique which leads to the differentiability of the value function. Therefore, the value function satisfies the corresponding Hamilton–Jacobi–Bellman equation in the classical sense, and the optimal control admits a state feedback representation. Under some additional conditions, it is shown that the value function is actually twice differentiable and the so-called quasi-Riccati equation is derived, whose solution can be used to construct the state feedback representation for the optimal control.

In this paper, we investigate the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid whose flow is assumed to be irrotational. Taking as control input the flow of the fluid through a part of the boundary of the rigid body, we obtain a finite-dimensional system similar to Kirchhoff laws in which the control input appears through both linear terms (with time derivative) and bilinear terms. Applying Coron’s return method, we establish some local controllability results for the position and velocities of the underwater vehicle. Examples with six, four, or only three controls inputs are given for a vehicle with an ellipsoidal shape.

In this article, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem.

The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of Γ-convergence, in the framework of finite plasticity. Denoting by ε the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order ε2α−2, with α ≥ 3. According to the value of α, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized plate theory.

We consider the problem of minimizing \hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathit{\ell }}\sqrt{{\mathit{\xi }}^{\mathrm{2}}\mathrm{+}{\mathit{K}}^{\mathrm{2}}\mathrm{\left(}\mathit{s}\mathrm{\right)}}\hspace{0.17em}\mathrm{d}\mathit{s}$ for a planar curve having fixed initial and final positions and directions. The total length ℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.

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].

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 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, 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.

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.

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. Illustrative examples show that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the stability analysis and the solution of robust feedback stabilization problems.

In this paper, we consider the well-known Fattorini’s criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686–694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini’s criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini’s criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier−Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini’s criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.