Gilson do Nascimento Silva, R. Andreani, Rui Marques Carvalho, L. Secchin
In this paper, we focus on quasi-Newton methods to solve constrained generalized equations.As is well-known, this problem was firstly studied by Robinson and Josephy in the 70's. Since then, it has been extensively studied by many other researchers, specially Dontchev and Rockafellar. Here, we propose two Broyden-type quasi-Newton approaches to dealing with constrained generalized equations, one that requires the exact resolution of the subproblems, and other that allows inexactness, which is closer to numerical reality. In both cases, projections onto the feasible set are also inexact. The local convergence of general quasi-Newton approaches is established under a bounded deterioration of the update matrix and Lipschitz continuity hypotheses. In particular, we prove that a general scheme converges linearly to the solution under suitable assumptions. Furthermore, when a Broyden-type update rule is used, the convergence is superlinearly. Some numerical examples illustrate the applicability of the proposed methods.
{"title":"Convergence of quasi-Newton methods for solving constrained generalized equations\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 ","authors":"Gilson do Nascimento Silva, R. Andreani, Rui Marques Carvalho, L. Secchin","doi":"10.1051/cocv/2022026","DOIUrl":"https://doi.org/10.1051/cocv/2022026","url":null,"abstract":"In this paper, we focus on quasi-Newton methods to solve constrained generalized equations.As is well-known, this problem was firstly studied by Robinson and Josephy in the 70's.\u0000Since then, it has been extensively studied by many other researchers, specially Dontchev and Rockafellar.\u0000Here, we propose two Broyden-type quasi-Newton approaches to dealing with constrained generalized equations, one that requires the exact resolution of the subproblems, and other that allows inexactness, which is closer to numerical reality.\u0000In both cases, projections onto the feasible set are also inexact.\u0000The local convergence of general quasi-Newton approaches is established under a bounded deterioration of the update matrix and Lipschitz continuity hypotheses.\u0000In particular, we prove that a general scheme converges linearly to the solution under suitable assumptions.\u0000Furthermore, when a Broyden-type update rule is used, the convergence is superlinearly.\u0000Some numerical examples illustrate the applicability of the proposed methods.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"96 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81159778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H^1(Omega) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar, we show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.
{"title":"Sensitivity analysis of a Tresca-type problem leads to Signorini's conditions","authors":"S. Adly, L. Bourdin, F. Caubet","doi":"10.1051/cocv/2022025","DOIUrl":"https://doi.org/10.1051/cocv/2022025","url":null,"abstract":"The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H^1(Omega) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar,\u0000we show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"8 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82447257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a class of particle systems which appear in various applications such as approximation theory, plasticity, potential theory and space-filling designs. The positions of the particles on the real line are described as a global minimum of an interaction energy, which consists of a nonlocal, repulsive interaction part and a confining part. Motivated by the applications, we cover non-standard scenarios in which the confining potential weakens as the number of particles increases. This results in a large area over which the particles spread out. Our aim is to approximate the particle interaction energy by a corresponding continuum interacting energy. Our main results are bounds on the corresponding energy difference and on the difference between the related potential values. We demonstrate that these bounds are useful to problems in approximation theory and plasticity. The proof of these bounds relies on convexity assumptions on the interaction and confining potentials. It combines recent advances in the literature with a new upper bound on the minimizer of the continuum interaction energy.
{"title":"Convergence rates for energies of interacting particles whose distribution spreads out as their number increases","authors":"P. Meurs, Ken’ichiro Tanaka","doi":"10.1051/cocv/2022083","DOIUrl":"https://doi.org/10.1051/cocv/2022083","url":null,"abstract":"We consider a class of particle systems which appear in various applications such as approximation theory, plasticity, potential theory and space-filling designs. The positions of the particles on the real line are described as a global minimum of an interaction energy, which consists of a nonlocal, repulsive interaction part and a confining part. Motivated by the applications, we cover non-standard scenarios in which the confining potential weakens as the number of particles increases. This results in a large area over which the particles spread out. Our aim is to approximate the particle interaction energy by a corresponding continuum interacting energy. Our main results are bounds on the corresponding energy difference and on the difference between the related potential values. We demonstrate that these bounds are useful to problems in approximation theory and plasticity. The proof of these bounds relies on convexity assumptions on the interaction and confining potentials. It combines recent advances in the literature with a new upper bound on the minimizer of the continuum interaction energy.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"4 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81942922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The ability to asymptotically stabilize control systems through the use of continuous feedbacks is an important topic of control theory and applications. In this paper, we provide a complete characterization of continuous feedback stabilizability using a new approach that does not involve control Lyapunov functions. To do so, we first develop a slight generalization of feedback stabilization using composition operators and characterize continuous stabilizability in this expanded setting. Employing the obtained characterizations in the more general context, we establish relationships between continuous stabilizability in the conventional sense and in the generalized composition operator sense. This connection allows us to show that the continuous stabilizability of a control system is equivalent to the stability of an associated system formed from a local section of the vector field inducing the control system. That is, we reduce the question of continuous stabilizability to that of stability. Moreover, we provide a universal formula describing all possible continuous stabilizing feedbacks for a given system.
{"title":"Continuous feedback stabilization of nonlinear control systems by composition operators","authors":"Bryce Chistopherson, B. Mordukhovich, F. Jafari","doi":"10.1051/cocv/2022022","DOIUrl":"https://doi.org/10.1051/cocv/2022022","url":null,"abstract":"The ability to asymptotically stabilize control systems through the use of continuous feedbacks is an important topic of control theory and applications. In this paper, we provide a complete characterization of continuous feedback stabilizability using a new approach that does not involve control Lyapunov functions. To do so, we first develop a slight generalization of feedback stabilization using composition operators and characterize continuous stabilizability in this expanded setting. Employing the obtained characterizations in the more general context, we establish relationships between continuous stabilizability in the conventional sense and in the generalized composition operator sense. This connection allows us to show that the continuous stabilizability of a control system is equivalent to the stability of an associated system formed from a local section of the vector field inducing the control system. That is, we reduce the question of continuous stabilizability to that of stability. Moreover, we provide a universal formula describing all possible continuous stabilizing feedbacks for a given system.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"36 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77439252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a $Gamma$-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the $L^2$-strong topology of the minimizers of the original problem. Using again a $Gamma$-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in $mathbb{R}^2$.
{"title":"Optimal control of ensembles of dynamical systems","authors":"Alessandro Scagliotti","doi":"10.1051/cocv/2023011","DOIUrl":"https://doi.org/10.1051/cocv/2023011","url":null,"abstract":"In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a $Gamma$-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the $L^2$-strong topology of the minimizers of the original problem. Using again a $Gamma$-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in $mathbb{R}^2$.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"72 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79208728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in [12] and settles a question emerged in [15]. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.
{"title":"The Sard problem in step 2 and in filiform Carnot groups\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 ","authors":"Francesco Boarotto, Luca Nalon, D. Vittone","doi":"10.1051/cocv/2022074","DOIUrl":"https://doi.org/10.1051/cocv/2022074","url":null,"abstract":"Abstract. We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in [12] and settles a question emerged in [15]. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"13 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84256662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Output-to-State Stability (OSS) is a notion of detectability for nonlinear systems that is formulated in the ISS framework. We generalize the notion of OSS for systems which possess a decomposable invariant set and evolve on compact manifolds. Building upon a recent extension of the ISS theory for this very class of systems [2], the paper provides equivalent characterizations of the OSS property in terms of asymptotic estimates of the state trajectories and, in particular, in terms of existence of smooth Lyapunov-like functions.
{"title":"Smooth output-to-state stability for multistable systems on compact manifolds","authors":"D. Angeli, Paolo Forni","doi":"10.1051/cocv/2022021","DOIUrl":"https://doi.org/10.1051/cocv/2022021","url":null,"abstract":"Output-to-State Stability (OSS) is a notion of detectability for nonlinear systems that is formulated in the ISS framework. We generalize the notion of OSS for systems which possess a decomposable invariant set and evolve on compact manifolds. Building upon a recent extension of the ISS theory for this very class of systems [2], the paper provides equivalent characterizations of the OSS property in terms of asymptotic estimates of the state trajectories and, in particular, in terms of existence of smooth Lyapunov-like functions.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"29 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77168974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on SL (2,R). Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in the literature We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups to identify exactly automorphic structures on which our results apply.
{"title":"Sub-Riemannian Geodesics on SL(2,R)","authors":"D. D’Alessandro, Gunhee Cho","doi":"10.1051/cocv/2022068","DOIUrl":"https://doi.org/10.1051/cocv/2022068","url":null,"abstract":"We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on SL (2,R). Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in the literature We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups to identify exactly automorphic structures on which our results apply.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"126 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73313610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is devoted to the long-standing problem on the smoothness of sub-Riemannian geodesics. We prove that the derivatives of sub-Riemannian geodesics are always $L_p$-H"older continuous. Additionally, this result has several interesting implications. These include (i) the decay of Fourier coefficients on abnormal controls, (ii) the rate at which they can be approximated by smooth functions, (iii) a generalization of the Poincar'e inequality, and (iv) a compact embedding of the set of shortest paths into the space of Bessel potentials.
{"title":"Derivatives of sub-Riemannian geodesics are $L_p$-Hölder continuous","authors":"L. Lokutsievskiy, M. Zelikin","doi":"10.1051/cocv/2023055","DOIUrl":"https://doi.org/10.1051/cocv/2023055","url":null,"abstract":"This article is devoted to the long-standing problem on the smoothness of sub-Riemannian geodesics. We prove that the derivatives of sub-Riemannian geodesics are always $L_p$-H\"older continuous. Additionally, this result has several interesting implications. These include (i) the decay of Fourier coefficients on abnormal controls, (ii) the rate at which they can be approximated by smooth functions, (iii) a generalization of the Poincar'e inequality, and (iv) a compact embedding of the set of shortest paths into the space of Bessel potentials.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"2019 39","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72400125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A general backward stochastic linear-quadratic optimal control problem is studied, in which both the state equation and cost functional contain the nonhomogeneous terms. The main feature of the problem is that the weighting matrices in the cost functional are allowed to be indefinite and cross-product terms in the control and the state processes are present. Necessary and sufficient conditions for the solvability of the problem are obtained, and a characterization of the optimal control in terms of forward-backward stochastic differential equations is derived. By a Riccati equation approach, a general procedure for constructing optimal controls is developed and the value function is obtained explicitly.
{"title":"General indefinite backward stochastic linear-quadratic optimal control problems","authors":"Jingrui Sun, Jiaqiang Wen, J. Xiong","doi":"10.1051/cocv/2022030","DOIUrl":"https://doi.org/10.1051/cocv/2022030","url":null,"abstract":"A general backward stochastic linear-quadratic optimal control problem is studied, in which both the state equation and cost functional contain the nonhomogeneous terms. The main feature of the problem is that the weighting matrices in the cost functional are allowed to be indefinite and cross-product terms in the control and the state processes are present. Necessary and sufficient conditions for the solvability of the problem are obtained, and a characterization of the optimal control in terms of forward-backward stochastic differential equations is derived. By a Riccati equation approach, a general procedure for constructing optimal controls is developed and the value function is obtained explicitly.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"37 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80107883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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