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We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately o...
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In the present paper, we are concerned with a class of constrained vector optimization problems, where the objective functions and active constraint functions are locally Lipschitz at the referee point. Some second-order constraint qualifications of Zangwill type, Abadie type and Mangasarian–Fromovitz type as well as a regularity condition of Abadi...
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... for all 0 < t ≤ s < T , where K is a positive constant that depends only on C. Next, we observe some classical properties on the relation between solutions of the continuity equation (8) and an associated ODE. This result enables some control on the growth of the support of the solution of the continuity equation, due to the Caratheodory existence theorem for solutions of ODEs. ...
... If r, R, C > 0 are such that supp µ 0 ⊂ B r (0), |V t (x)| < C for all (t, x) ∈ [0, T ] × B R+r (0) and T < R+r C . Then there exists a unique solution µ to the continuity equation (8). Additionally, the solution µ is given by µ ...
... Therefore, there exists a subsequence of {µ N } N ∈Z+ that converges to a limitμ in C([0, T ]; P 2 (R d )). Next, we will verify thatμ is the weak solution of the continuity equation (8) corresponding to the curve V . Let φ ∈ C ∞ ([0, T ) × R d ) be a compactly supported function. ...
We consider the controllability problem for the continuity equation, corresponding to neural ordinary differential equations (ODEs), which describes how a probability measure is pushedforward by the flow. We show that the controlled continuity equation has very strong controllability properties. Particularly, a given solution of the continuity equation corresponding to a bounded Lipschitz vector field defines a trajectory on the set of probability measures. For this trajectory, we show that there exist piecewise constant training weights for a neural ODE such that the solution of the continuity equation corresponding to the neural ODE is arbitrarily close to it. As a corollary to this result, we establish that the continuity equation of the neural ODE is approximately controllable on the set of compactly supported probability measures that are absolutely continuous with respect to the Lebesgue measure.
... Theorem 1 (Main Result): Suppose that Assumptions 2 and 1 hold and μ 0 ∈ P 2 (R d ) has compact support. Let μ be the weak solution of the continuity equation (8) corresponding to the vector field V. Additionally, suppose that V is uniformly bounded in space and time. Then for every > 0, there exist piecewise constant control inputs A (·), W (·) and θ (·), such that the corresponding weak solutions μ of (4) satisfy ...
... Next, we observe some classical properties on the relation between solutions of the continuity equation (8) and an associated ODE. This result enables some control on the growth of the support of the solution of the continuity equation, due to the Caratheodory existence theorem for solutions of ODEs. ...
... If r, R, C > 0 are such that supp μ 0 ⊂ B r (0), |V t (x)| < C for all (t, x) ∈ [0, T] × B R+r (0) and T < R+r C . Then there exists a unique solution μ to the continuity equation (8). Additionally, the solution μ is given ...
We consider the controllability problem for the continuity equation, corresponding to neural ordinary differential equations (ODEs), which describes how a probability measure is pushedforward by the flow. We show that the controlled continuity equation has very strong controllability properties. Particularly, a given solution of the continuity equation corresponding to a bounded Lipschitz vector field defines a trajectory on the set of probability measures. For this trajectory, we show that there exist piecewise constant training weights for a neural ODE such that the solution of the continuity equation corresponding to the neural ODE is arbitrarily close to it. As a corollary to this result, we establish that the continuity equation of the neural ODE is approximately controllable on the set of compactly supported probability measures that are absolutely continuous with respect to the Lebesgue measure.
... Beside the dynamics of mean-field equations, it is now relevant to study control problems for them, that are known as mean-field control problems. In the mean-field limit for deterministic models, a few articles have been dealing with controllability results [29,30] or explicit syntheses of control laws [18,44]. Most of the literature focused on optimal control problems, with contributions ranging from existence results [15,31,32,33] to first-order optimality conditions [7,11,12,13,14,23,24,45], to numerical methods [1,16]. ...
We show the existence of Lipschitz-in-space optimal controls for a class of mean-field control problems with dynamics given by a non-local continuity equation. The proof relies on a vanishing viscosity method: we prove the convergence of the same problem where a diffusion term is added, with a small viscosity parameter. By using stochastic optimal control, we first show the existence of a sequence of optimal controls for the problem with diffusion. We then build the optimizer of the original problem by letting the viscosity parameter go to zero.
... This family of models refers broadly to situations in which a centralised policy-making entity emits a control signal at the macroscopic level, in order to stir an underlying microscopic multi-agent system towards a desired goal (see the introduction of [20] for more details on this general scheme). While a few results have been dealing in this context with controllability issues [49,50] as well as the explicit synthesis of control laws for consensus and alignment models [11,29,30,75], the core of the existing contributions on this topic pertains to mean-field optimal control. In this setting, a first series of articles have aimed at studying rigorously the mean-field limit of solutions of optimal control problems formulated on discrete particle-like systems [38,53,54,55]. ...
In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems.
... If the measure is absolutely continuous with respect to the Lebesgue measure, is it true that time modulations of the flow (1.2) do not concentrate mass along the boundary, then eventually giving a non-zero probability of being close to unsafe configurations? This problem has been addressed in [8,9]. This is also one of the main motivations and future applications of the result presented here: to develop a theory describing the reachable set starting from a measure under control action. ...
The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e. its Lebesgue measure is zero). We also provide several counterexample showing that the hypotheses of our theorem are close to sharp.
... Berman et al. (2009);Elamvazhuthi et al. (2018); Elamvazhuthi and Berman (2019)). A few articles have been dealing with controllability results (see Duprez et al. (2019Duprez et al. ( , 2020) or explicit syntheses of control laws (e.g. Caponigro et al. (2015); Piccoli et al. (2015)). ...
We study a family of optimal control problems in which one aims at minimizing a cost that mixes a quadratic control penalization and the variance of the system, both for finite sets of agents and for the limit problem as their number goes to infinity. While the solution for finitely many agents always exists in a unique and explicit form, the behavior of the corresponding limit problem is very sensitive to the magnitude of the time horizon and penalization parameter. For the variance minimization, there exist Lipschitz-in-space optimal controls for the infinite dimensional problem, which can be obtained as a suitable limit of the optimal controls for the finite-dimensional problems. The same holds true for the variance maximization with a sufficiently small final time. Instead, for large final times (or equivalently large penalizations of the variance), Lipschitz-regular optimal controls do not exist for the macroscopic problem.
... The results in Section 6, which provide a transport formulation of Deep Learning, making use of the classical link between ODEs and transport equations, can be seen as bilinear-type simultaneous control results for Neural transport equations and systems (Theorems 5 and 6). The bilinear control of transport equations has also been considered (see, for instance, [12,13]) with controls localized in space motivated by collective behavior problems [28]. The nature of the controls we employ here and the dynamics that the systems under consideration, exploiting the very features of activation functions, are completely different. ...
We analyze Neural Ordinary Differential Equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of Deep Learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows achieving our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.
... The controllability of the continuity equation, i.e. System (1.1) without diffusion, has been investigated in [18,19]. ...
... [3]), an important research effort has been directed towards the generalisation of tools of control theory to the metric setting of the space of probability measures. The corresponding contributions include controllability results [19], existence of optimal controls [8,20,22], optimality conditions [5,6,7,25] and numerical methods [11]. ...
... where U (µ, ν M ) is defined as in (19) by identifying the empirical measure ν M with the set of points y ∈ (R d ) M where it is supported. It can be verified that all the objects involved in (P ) satisfy hypotheses (OCP) of Section 4. Therefore, problem (P ) has an optimal solution. ...
In this article, we propose a new unifying framework for the investigation of multi-agent control problems in the mean-field setting. Our approach is based on a new definition of differential inclusions for continuity equations formulated in the Wasserstein spaces of optimal transport. The latter allows to extend several known results of the classical theory of differential inclusions, and to prove an exact correspondence between solutions of differential inclusions and control systems. We show its appropriateness on an example of leader-follower evacuation problem.
... During the last few years, an important research effort at the interface between control theory and calculus of variations has been directed towards control problems formulated on continuity equations of the form (2). While a few results have been dealing with controllability properties of continuity equations [36,37], the major part of the literature has been devoted to the study of optimal control problems, with contributions ranging from existence results [23,38,39,40] and necessary optimality conditions [1,21,22,24,27,29,31,56] to numerical methods [28,57]. All these findings have hugely benefited from theoretical progresses made in the theory of optimal transport, for which we refer largely to the reference monographs [10,58,59]. ...
In this article, we propose a general framework for the study of differential inclusions in the Wasserstein space of probability measures. Based on earlier geometric insights on the structure of continuity equations, we define solutions of differential inclusions as absolutely continuous curves whose driving velocity fields are measurable selections of multifunction taking their values in the space of vector fields. In this general setting, we prove three of the founding results of the theory of differential inclusions: Filippov's theorem, the Relaxation theorem, and the compactness of the solution sets. These contributions -- which are based on novel estimates on solutions of continuity equations -- are then applied to derive a new existence result for fully non-linear mean-field optimal control problems with closed-loop controls.
