This work addresses the controllability of the energy of quantum bounded states through domain deformations in the two-dimensional framework. We approach the controllability question from a practical point of view, with the primary goal of providing simple and implementable control processes involving moving rectangles. We numerically validate the feasibility of our controls and analyze the energy transitions that occur in quantum states confined within specific deformations. This work presents two possible approaches for numerically implementing domain deformations. We show that one can either apply a suitable change of variables to fix the domain, or simulate the deformation using high-intensity confining potentials.
In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in a two-dimensional bounded domain. We formulate a distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier-Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite-dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via the comparison principle.
This paper studies the feedback stabilizability of linear abstract control systems with unbounded control operators, whose state operators generate analytic semigroups. Under an almost exponential decay condition that ensures the existence of bounded feedback operators, we establish several equivalent characterizations of stabilizability. Our main tool is an extended-state-space construction in which the control operator becomes admissible, combined with a Hautus-type resolvent (frequency-domain) inequality. As an application, we recover and extend existing results to settings in which the semigroup generated by the state operator is non-compact and the control operator may fail to be admissible.
This article examines a well-known analogue concept of null controllability in a discrete setting, referred to as φ-null controllability, specifically for a fully discrete approximation of the linear Korteweg-de Vries (KdV) equation using a single boundary control. To conduct this study, we first derive a suitable Carleman estimate for a fully discrete approximation of the differential operator Bt`Bx3. We then use this discrete Carleman estimate to deduce a uniform discrete observability estimate satisfied by the solution of fully-discrete adjoint system associated to the discrete control system. Finally, we illustrate how this uniform discrete observability inequality satisfied by the solution of discrete adjoint system establishes the φ-null controllability of the underlying discrete control system.
Over the last years, minimization problems over spaces of measures have received increased interest due to their relevance in the context of inverse problems, optimal control and machine learning. A fundamental role in their numerical analysis is played by the assumption that the optimal dual state admits finitely many global extrema and satisfies a second-order sufficient optimality condition in each one of them. In this work, we show the full equivalence of these structural assumptions to a no-gap second-order condition involving the second subderivative of the Radon norm as well as to a local quadratic growth property of the objective functional with respect to the bounded Lipschitz norm.
Necessary first-order conditions for a strong local minimum in the simplest problem of the calculus of variations, as well as in the Mayer-type optimal control problem with a control constraint u is an element of U are discussed. In the simplest problem, these conditions pertain to the break point of the extremal, and in the Mayer problem, to the control discontinuity point. Like the classical Weierstrass-Erdmann conditions, our conditions follow from the Weierstrass condition (respectively, from the Pontryagin maximum principle). In the simplest problem, a piecewise smooth extremal with a single discontinuity point of the derivative is considered. According to the Weierstrass-Erdmann conditions, a jump in the derivative is possible only if a non-strict inequality in the Weierstrass condition is satisfied, namely, at the point where this inequality is realized as an equality. The same is true for a control jump and the maximum principle. All possible cases of this situation are analyzed, including "control jumps" at the ends of the time interval. The obtained conditions are new both for the calculus of variations and for optimal control.
A relaxation problem for maps from 3-dimensional domains into the unit 2-sphere is analyzed, the energy being given in the smooth case by the integral of the modulus of the Laplacean vector. For second order Sobolev maps, a complete explicit formula of the relaxed energy is obtained. Our proof is based on the following results: minimal energy computation of maps with fixed degree, dipole-like problems, lower semicontinuity of the extended energy, and a strong approximation result on Cartesian currents.
We formulate and investigate a convex optimization problem defined on a set of probability measures mu with prescribed marginal m, which we call Mean Field Optimization (MFO) problem. The cost function depends on an aggregate term, defined as the expectation of mu with respect to a contribution function. This problem is of particular interest in the context of Lagrangian potential mean field games and their discretization. We provide a first-order optimality condition and prove strong duality. We investigate stability properties of the MFO problem with respect to the prescribed marginal, from both primal and dual perspectives. In our stability analysis, we propose a method for recovering an approximate solution to an MFO problem with the help of an approximate solution to an MFO with a different marginal m, typically an empirical distribution. We combine this method with the stochastic Frank-Wolfe algorithm of [Bonnans et al. SIAM J. Optim.33 (2023) 3083-3113] to derive a complete resolution method.
We consider an optimal control problem for the heat equation as a prototypical parabolic partial differential equation with a non-convex control mechanism of the form continuous-or-off. We model this fundamental switching mechanism as the product of a classically continuous and a binary control both in the control term of the dynamics and in the objective. A total variation regularization is added to the cost in order to restrict the number of switching times. This leads to a mixed-integer non-linear PDE-constrained problem. We discuss well-posedness of the problem and present an exact relaxation result for a linearized and a trust-region type penalized problem. The exactness result is constructive and provides a way to numerically compute mixed-integer optimal solutions from the optimality conditions of an associated PDE-constrained problem without integer restrictions. It lays a foundation for a new class of sequential relaxation algorithms to solve the considered class of mixed-integer control problems. This is demonstrated numerically by showcasing a descent step in the presence of binary restrictions.
We study directional differentiability properties of solution operators of rate-independentevolution variational inequalities with full-dimensional convex polyhedral admissible sets. It is shownthat, if the space of continuous functions of bounded variation is used as the domain of definition, thenthe most prototypical examples of such solution operators - the vector play and stop - are Hadamarddirectionally differentiable in a pointwise mannerif and only ifthe admissible set is non-obtuse. Wefurther prove that, in those cases where they exist, the directional derivatives of the vector play andstop are uniquely characterized by a system of projection identities and variational inequalities andthat directional differentiability cannot be expected in the obtuse case even if the solution operatoris restricted to the space of Lipschitz continuous functions. Our results can be used, for example, toformulate Bouligand stationarity conditions for optimal control problems involving sweeping processes.
We consider a class of infinite-dimensional linear control systems with controls subject to possible saturation. The aim is to study the stabilization under the additional switching control constraint which means that only one actuator is active. The appropriate feedback turns out to be multivalued and the existence of the solution to the resulting differential inclusion is established by using nonlinear semigroup theory. The switching control property occurs whenever the set of instants at which the trajectory intersects some specified subsets of the state space is of null measure. In the multivalued feedback framework, asymptotic stability and asymptotic output stability are obtained from LaSalle invariance principle. We establish also other stabilization results by constructing suitable state dependent switched systems where only one control is activated in each subsystem. Applications to the simultaneous stabilization under switching control constraint are treated for various partial differential equations. The latter includes heat, plate and wave systems.
In recent years, a novel pole placement paradigm has emerged for linear time-invariant systems of functional differential equations, addressing the complexities posed by their infinite spectrum and intricate dynamics. This paradigm is built upon the multiplicity-induced-dominancy (MID) property, which asserts that a characteristic root with sufficiently high multiplicity can dominate the spectral behavior of the system, thereby influencing its dynamic response. While the MID property has proven to be a powerful tool in control design, its applicability often depends on specific system configurations and parameter constraints. In this study, we propose new sufficient conditions ensuring the validity of the MID property in the context of the lowest intermediate over-order multiplicity. Our approach establishes a connection between the MID property and Laguerre polynomials by exploiting their inherent properties. Thanks to their structural properties, these conditions are tailored to the prescribed stabilization of conservative mechanical systems, which are characterized in the Laplace domain by even polynomials. By leveraging these conditions, we present a systematic framework for the precise assignment of dominant roots in the spectrum of conservative mechanical systems stabilized via delayed state feedback, enabling accurate control over both the system's solution's long-time behavior as well as its exact exponential decay.
The magnetization control problem for the Landau-Lifshitz-Gilbert (LLG) equation mt = m & times; (Delta m + u) - m & times; (m & times; (Delta m + u)), (x, t) is an element of Omega & times; (0, T] with zero Neumann boundary data on a two-dimensional bounded domain Omega is studied when the control energy u is applied on the effective field. First, we show the existence of a weak solution, and that the magnetization vector field m satisfies an energy inequality. If a weak solution m obeys the condition that del m is an element of L4(0,T; L4(Omega)), then we show that it is a regular solution. The classical cost functional is modified by incorporating L4(0, T; L4(Omega))-norm of del m, enabling a rigorous study of the optimal control problem. Then, we justify the existence of an optimal control and derive first-order necessary optimality conditions using an adjoint problem approach. We establish the continuous dependency and Fr & eacute;chet differentiability of the control-to-state and control-to-costate operators and show the locally Lipschitz continuity of their Frechet derivatives. Using these postulates, we derive a local second-order sufficient optimality condition. Moreover, we modified the control problem and derived a global optimality criterion. Finally, we obtain another remarkable second-order sufficient optimality condition along a cone of critical directions where the control stems from a finite number of fixed magnetic field coils.
In this paper, we study an optimal control problem of partially observed stochastic evolution equation control system. The observation process is an infinite-dimensional stochastic process, and the control variable is allowed to enter the diffusion term of the state process and the drift term of the observation process. The control domain need not be convex. Combined with Girsanov theorem, we establish the related stochastic maximum principle in the sense of transposition solution.
This paper investigates the turnpike properties of deterministic nonzero-sum linearquadratic (LQ) differential games. Under certain assumptions on the Hamiltonian matrix of the nonzero-sum LQ differential game, we establish the solvability of both the coupled non-symmetric differential Riccati equation (DRE) and the algebraic Riccati equation (ARE). Moreover, we identify the convergence relationship between the DRE and ARE, which is essential for understanding the turnpike properties. Over a finite but sufficiently long time horizon, the open-loop Nash equilibrium is shown to remain exponentially close to the solution of a two-objective optimization problem for the majority of the time horizon.
In this paper, we focus on the Schrödinger equations with inverse-square potentials in dimension one; these special potentials play an important role in the field of mathematical physics. We study several observability and unique continuation inequalities at one time point or at two time points for these equations. These observability and unique continuation inequalities are some new types of quantitative estimates which have appeared in recent literature. Their proofs essentially rely on the representation of the solution, a Nazarov-type uncertainty principle for the Hankel transform, and an interpolation inequality for functions whose Hankel transforms have compact support. Meanwhile, these inequalities can be applied to the controllability of these Schrödinger equations.
This paper presents an innovative probabilistic control framework for continuous-time stochastic systems. Unlike traditional control approaches that optimise deterministic control strategies, our framework directly optimises the probability density function (PDF) of the control signal, allowing for a more adaptable and robust response to stochastic variations. By integrating stochastic differential equations with the Hamilton–Jacobi–Bellman equation and utilising the Fokker–Planck dynamics, our method offers a precise and dynamic approach to managing uncertainty. The framework minimises the Kullback–Leibler divergence to align the system’s joint state and control distribution with a desired joint target distribution, ensuring effective control even in unpredictable environments. A novel algorithm iteratively refines the control PDF based on real-time feedback, further enhancing the system’s alignment with the target behaviour. The proposed method is demonstrated on an Ornstein–Uhlenbeck process, showcasing its effectiveness in steering the system’s state distribution toward desired outcomes and underscoring its broad applicability to stochastic systems.
This paper provides two general representations of the weak limit u(t,x) of the solution uε(t,x) for (t,x) ∈ (0,T)×Rd, to the linear transport equation with an oscillating regular velocity b(x/ε), an initial datum u0ε(x) and a right-hand side fε(t,x). Our main assumption is the existence of a function w ∈ C1(Rd) such that b · ∇w is bounded from below by a positive constant. As a consequence, the dynamic flow Φ(t,y) associated with the vector field b(y) induces a one-to-one mapping from R×Σ onto Rd, where Σ is the equipotential hypersurface {w = 0}. This assumption allows us to finely characterize the kernel of the differential operator b(y) · ∇y(·) thanks to some quotient set Σb/R, where Σb is a suitable compact subset of Σ. Then, using a two-scale procedure we establish two integral formulas of the limit u(t,x), one over the quotient set and the other one over the d-dimensional torus Td, which involve the two-scales limits of the sequences of data and the projection of b onto the kernel of b(y) · ∇y(·). Finally, we also derive two alternative expressions of the asymptotics of the flow lim∞ Φ(t,y)/t for a.e. y ∈ Td.
The main aim of this paper is to develop a theory for non-autonomous parabolic equations with time-dependent measures on the spatial domain appearing as right hand sides. Restricting these measures to ones which have their supports on 'curves' or 'surfaces' - the latter understood in the sense of geometric measure theory - we succeed in interpreting them as distributional objects from a (negative indexed) Sobolev-Slobodetskii space Ws,2(Omega) with s close to -1. For these indices s a tailor suited parabolic theory is established, based on results of Disser et al. [Ann. Sci. Norm. Super. Pisa, Cl. Sci. 17 (2017) 65-79] and Haller-Dintelmann et al. [Ann. Mat. Pura Appl. 198 (2019) 1227-1241]. The proposed frame work is well-suited for optimal control problems with controls acting on sub-manifolds.
In this paper, we study the following fractional Schro ̈dinger equation (−∆)su+λV(x)u = ulogu2, x ∈ RN, where 0 < s < 1, λ > 0 and V(x) : RN → R is a measurable potential satisfying some assumptions. Since the logarithmic term is singular at the origin, the energy functional is not of class C1. To overcome this difficulty, the nonsmooth critical point theory developed by Szulkin is used. Employing variational methods, the existence of a positive least energy solution for the problem is established for sufficient large λ’s. In addition, we prove that such solutions converge to a least energy solution of the limit problem defined on a bounded domain as λ → ∞.
