Papers by Giovanni Colombo
Journal of Differential Equations, 2021
The paper is devoted to deriving necessary optimality conditions in a general optimal control pro... more The paper is devoted to deriving necessary optimality conditions in a general optimal control problem for dynamical systems governed by controlled sweeping processes with hard-constrained control actions entering both polyhedral moving sets and additive perturbations. By using the first-order and mainly second-order tools of variational analysis and generalized differentiation, we develop a well-posed method of discrete approximations, obtain optimality conditions for solutions to discrete-time control systems, and then establish by passing to the limit verifiable necessary optimality conditions for local minimizers of the original controlled sweeping process that are expressed entirely in terms of its given data. The efficiency of the obtained necessary optimality conditions for the sweeping dynamics is illustrated by solving three nontrivial examples of their own interest.
Discrete & Continuous Dynamical Systems, 2021
We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyh... more We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger \begin{document}$ W^{1,2} $\end{document} convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.
Mathematics of Operations Research, Aug 1, 2013
, https://www.math.lsu.edu/~wolenski/ We consider the class of continuous functions that map an o... more , https://www.math.lsu.edu/~wolenski/ We consider the class of continuous functions that map an open set ⊆ n to with an epigraph having (locally) positive reach with an additional property. This class contains all finite convex and C 1 1 functions, but also ones that are not necessarily Lipschitz continuous. We provide a representation formula for the Clarke generalized gradient of such functions using convex combinations and limits of gradients at differentiability points, thus offering an alternative to the well-known proximal normal formula by replacing a pointedness assumption by one of positive reach. Our proof consists of a detailed analysis of singularities using methods taken from both nonsmooth analysis and geometric measure theory, and is based on an induction argument. As an application, we prove for a particular class of Hamilton-Jacobi equations that an a.e. solution whose hypograph has positive reach and satisfies an additional property is indeed the unique viscosity solution.
Differentiability properties for a class of non-convex functions
Calculus of Variations and Partial Differential Equations, Oct 28, 2005
Abstract Closed sets K⊂\ mathbb R^ n satisfying an external sphere condition with uniform radius ... more Abstract Closed sets K⊂\ mathbb R^ n satisfying an external sphere condition with uniform radius (called ϕ-convexity or proximal smoothness) are considered. It is shown that for\ mathcal H^ n-1-ae x∊∂ K the proximal normal cone to K at x has dimension one. ...
Siam Journal on Control and Optimization, 2006
A minimal time problem with linear dynamics and convex target is considered. It is shown, essenti... more A minimal time problem with linear dynamics and convex target is considered. It is shown, essentially, that the epigraph of the minimal time function T (•) is ϕ-convex (i.e., it satisfies a kind of exterior sphere condition with locally uniform radius), provided T (•) is continuous. Several regularity properties are derived from results in [G. Colombo and A. Marigonda, Calc. Var. Partial Differential Equations, 25 (2005), pp. 1-31], including twice a.e. differentiability of T (•) and local estimates on the total variation of DT .
Non-Lipschitz points and the $${\textit{SBV}}$$ SBV regularity of the minimum time function
Calculus of Variations and Partial Differential Equations, Nov 12, 2013
This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum ... more This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function $$T$$T under controllability conditions which do not imply the Lipschitz continuity of $$T$$T. We consider first the case of normal linear control systems with constant coefficients in $${\mathbb {R}}^N$$RN. We characterize points around which $$T$$T is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call $$\mathcal {S}$$S. Furthermore, we show that $$\mathcal {S}$$S is countably $$\mathcal {H}^{N-1}$$HN-1-rectifiable with positive $$\mathcal {H}^{N-1}$$HN-1-measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set $$\mathcal {S}$$S in a neighborhood of the origin in a similar way and prove the $$\mathcal {H}^1$$H1-rectifiability of $$\mathcal {S}$$S and that $$\mathcal {H}^1(\mathcal {S})>0$$H1(S)>0. In both cases, $$T$$T is known to have epigraph with positive reach, hence to be a locally $$BV$$BV function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of $$DT$$DT must be concentrated in $$\mathcal {S}$$S, our analysis yields that $$T$$T is locally $$SBV$$SBV, i.e., the Cantor part of $$DT$$DT vanishes. Our results imply also that $$T$$T is differentiable outside a $$\mathcal {H}^{N-1}$$HN-1-rectifiable set. With small changes, our results are valid also in the case of multiple control input.
Springer INdAM Series, 2015
This paper is devoted to second-order necessary optimality conditions for the Mayer optimal contr... more This paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem with an arbitrary closed control set U ⊂ R m. Admissible controls are supposed to be measurable and essentially bounded. Using second order tangents to U , we first show that ifū(•) is an optimal control, then an associated quadratic functional should be nonnegative for all elements in the second order jets to U alongū(•). Then we specify the obtained results in the case when U is given by a finite number of C 2-smooth inequalities with positively independent gradients of active constraints. The novelty of our approach is due, on one hand, to the arbitrariness of U. On the other hand, the proofs we propose are quite straightforward and do not use embedding of the problem into a class of infinite dimensional mathematical programming type problems. As an application we derive new second-order necessary conditions for a free end-time optimal control problem in the case when an optimal control is piecewise Lipschitz.
We formulate and study an optimal control problem for the sweeping (Moreau) process, where contro... more We formulate and study an optimal control problem for the sweeping (Moreau) process, where control functions enter the moving sweeping set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and generalized differentiation. The final results obtained are given in terms of the initial data of the controlled sweeping process and are illustrated by nontrivial examples.
Approximate and relaxed solutions of differential inclusions
… del Seminario Matematico della Università di …, 1989
Journal of Differential Equations, 2016
The paper addresses a new class of optimal control problems governed by the dissipative and disco... more The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong W 1,2-convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples.
Singularities for a class of non-convex sets and functions, and viscosity solutions of some Hamilton-Jacobi equations
Journal of Convex Analysis
We study nondifferentiability points for a class of continuous functions f:ℝ N →ℝ whose epigraph ... more We study nondifferentiability points for a class of continuous functions f:ℝ N →ℝ whose epigraph satisfies a kind of external sphere condition with uniform radius (called φ-convexity or proximal smoothness). The functions belonging to this class are not necessarily Lipschitz. However, they enjoy some properties analogous to semiconvex functions; in particular they are twice ℒ N -a.e. differentiable (see the authors [Calc. Var. Partial Differ. Equ. 25, No. 1, 1–31 (2006; Zbl 1082.49018)]). In partial analogy with the study of singularities of semiconcave functions (see P. Cannarsa and C. Sinestrari [Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Boston, MA: Birkhäuser (2004; Zbl 1095.49003)]), under suitable conditions we give estimates from below of the nondifferentiability set, which consists of points where the subdifferential is not a singleton, as well as (differently from semiconvex functions) of points where it is empty. Furthermore, we show that if a f...
This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum ... more This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function T under controllability conditions which do not imply the Lipschitz continuity of T . We consider first the case of normal linear control systems with constant coefficients in R N . We characterize points around which T is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call S. Furthermore, we show that S is H N-1 -rectifiable with positive H N-1 -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set S in a neighborhood of the origin in a similar way and prove the H 1 -rectifiability of S and that H 1 (S) > 0. In both cases, T is known to have epigraph with positive reach, hence to be a locally BV function (see ). Since the Cantor part of DT must be concentrated in S, our analysis yields that T is SBV , i.e., the Cantor part of DT vanishes. Our results imply also that T is locally of class C 1,1 outside a H N-1 -rectifiable set. With small changes, our results are valid also in the case of multiple control input.
The paper is devoted to the study of a new class of optimal control problems governed by the clas... more The paper is devoted to the study of a new class of optimal control problems governed by the classical Moreau sweeping process with the new feature that the polyhedral moving set is not fixed while controlled by time-dependent functions. The dynamics of such problems is described by dissipative non-Lipschitzian differential inclusions with state constraints of equality and inequality types. It makes challenging and difficult their analysis and optimization. In this paper we establish some existence results for the sweeping process under consideration and develop the method of discrete approximations that allows us to strongly approximate, in the W 1,2 topology, optimal solutions of the continuous-type sweeping process by their discrete counterparts.
An Existence Result for Differential Inclusions with Non-Convex Right-Hand Side
Funkcialaj Ekvacioj, 1989
Consider a closed subset $K$ of $X=R^{n}$ and a continuous map $f:K¥rightarrow R^{n}$ , satisfyin... more Consider a closed subset $K$ of $X=R^{n}$ and a continuous map $f:K¥rightarrow R^{n}$ , satisfying the tangential condition ... (T) $¥lim_{h¥rightarrow}¥inf_{0}¥frac{d(x+hf(x),K)}{h}=0$ ... (1) $¥left¥{¥begin{array}{l}¥chi^{¥prime}=f(¥mathrm{x})¥¥x(0)=x_{0},¥end{array}¥right.$
Journal of Differential Equations
This paper addresses a new class of optimal control problems for perturbed sweeping processes wit... more This paper addresses a new class of optimal control problems for perturbed sweeping processes with measurable controls in additive perturbations of the dynamics and smooth controls in polyhedral moving sets. We develop a constructive discrete approximation procedure that allows us to strongly approximate any feasible trajectory of the controlled sweeping process by feasible discrete trajectories and also establish a W 1,2-strong convergence of optimal trajectories for discretized control problems to a given local minimizer of the original continuous-time sweeping control problem of the Bolza type. Employing advanced tools of first-order and secondorder variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete optimal solutions under fairly general assumptions formulated entirely in terms of the given data. The obtained results give us efficient suboptimality ("almost optimality") conditions for the original sweeping control problem that are illustrated by a nontrivial numerical example.
On the optimal control of rate-independent soft crawlers
Journal de Mathématiques Pures et Appliquées
Journal of Convex Analysis, 2004
Giovanni Colombo∗ Dipartimento di Matematica Pura e Applicata, Universit`a di Padova, via Belzoni... more Giovanni Colombo∗ Dipartimento di Matematica Pura e Applicata, Universit`a di Padova, via Belzoni 7, 35131 Padova, Italy [email protected] ... Peter R. Wolenski Department of Mathematics, Louisiana State University, 326 Lockett Hall, Baton Rouge, Louisiana ...
SIAM Journal on Control and Optimization
Let C(t), t ≥ 0 be a Lipschitz set-valued map with closed and (mildly non-)convex values and f (t... more Let C(t), t ≥ 0 be a Lipschitz set-valued map with closed and (mildly non-)convex values and f (t, x, u) be a map, Lipschitz continuous w.r.t. x. We consider the problem of reaching a target S within the graph of C subject to the differential inclusion
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Papers by Giovanni Colombo