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Sanjay Lall - One of the best experts on this subject based on the ideXlab platform.
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convexity of decentralized controller synthesis
IEEE Transactions on Automatic Control, 2016Co-Authors: Laurent Lessard, Sanjay LallAbstract:In decentralized control problems, a standard approach is to specify the set of allowable decentralized controllers as a Closed Subspace of linear operators. This then induces a corresponding set of Youla–Kucera parameters. Previous work has shown that quadratic invariance of the controller set implies that the set of Youla–Kucera parameters is convex. In this technical note, we prove the converse. We thereby show that the only decentralized control problems for which the set of Youla–Kucera parameters is convex are those which are quadratically invariant. We further show that under additional assumptions, quadratic invariance is necessary and sufficient for the set of achievable Closed-loop maps to be convex. We give two versions of our results. The first applies to bounded linear operators on a Banach space and the second applies to (possibly unstable) causal LTI systems in discrete or continuous time.
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convexity of decentralized controller synthesis
arXiv: Systems and Control, 2013Co-Authors: Laurent Lessard, Sanjay LallAbstract:In decentralized control problems, a standard approach is to specify the set of allowable decentralized controllers as a Closed Subspace of linear operators. This then induces a corresponding set of Youla parameters. Previous work has shown that quadratic invariance of the controller set implies that the set of Youla parameters is convex. In this paper, we prove the converse. We thereby show that the only decentralized control problems for which the set of Youla parameters is convex are those which are quadratically invariant. We further show that under additional assumptions, quadratic invariance is necessary and sufficient for the set of achievable Closed-loop maps to be convex. We give two versions of our results. The first applies to bounded linear operators on a Banach space and the second applies to (possibly unstable) causal LTI systems in discrete or continuous time.
Katrin Wehrheim - One of the best experts on this subject based on the ideXlab platform.
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space valued Cauchy-Riemann equations with totally real boundary conditions
2013Co-Authors: Katrin WehrheimAbstract:The main purpose of this paper is to give a general regularity result for Cauchy-Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic L p-regularity results hold true under one crucial assumption: The Banach space is isomorphic to a Closed Subspace of an L p-space. (Equivalently, the totally real submanifold is modelled on a Closed Subspace of an L p-space.) Some minor corrections are in order on the Sobolev arithmetic in the estimates. Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-manifolds with a boundary space-time splitting, leading towards the definition of a Floer homology for 3-manifolds with boundary, which is the first step in a program by Salamon for the proof of the Atiyah-Floer conjecture. The principal part of such a boundary value problem is an example of a Banach space valued Cauchy-Riemann equation with totally real boundary condition.
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banach space valued cauchy riemann equations with totally real boundary conditions
Communications in Contemporary Mathematics, 2004Co-Authors: Katrin WehrheimAbstract:The main purpose of this paper is to give a general regularity result for Cauchy–Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic Lp-regularity results hold true under one crucial assumption: The Banach space is isomorphic to a Closed Subspace of an Lp-space. (Equivalently, the totally real submanifold is modelled on a Closed Subspace of an Lp-space.) Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-manifolds with a boundary space-time splitting, leading towards the definition of a Floer homology for 3-manifolds with boundary, which is the first step in a program by Salamon for the proof of the Atiyah–Floer conjecture. The principal part of such a boundary value problem is an example of a Banach space valued Cauchy–Riemann equation with totally real boundary condition.
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banach space valued cauchy riemann equations with totally real boundary conditions
arXiv: Analysis of PDEs, 2004Co-Authors: Katrin WehrheimAbstract:The main purpose of this paper is to give a general regularity result for Cauchy-Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic $L^p$-regularity results hold true under one crucial assumption: The totally real submanifold has to be modelled on an $L^p$-space or a Closed Subspace thereof. Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-manifolds with a boundary space-time splitting, leading towards the definition of a Floer homology for 3-manifolds with boundary, which is the first step in a program by Salamon for the proof of the Atiyah-Floer conjecture. The principal part of such a boundary value problem is an example of a Banach space valued Cauchy-Riemann equation with totally real boundary condition.
Laurent Lessard - One of the best experts on this subject based on the ideXlab platform.
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convexity of decentralized controller synthesis
IEEE Transactions on Automatic Control, 2016Co-Authors: Laurent Lessard, Sanjay LallAbstract:In decentralized control problems, a standard approach is to specify the set of allowable decentralized controllers as a Closed Subspace of linear operators. This then induces a corresponding set of Youla–Kucera parameters. Previous work has shown that quadratic invariance of the controller set implies that the set of Youla–Kucera parameters is convex. In this technical note, we prove the converse. We thereby show that the only decentralized control problems for which the set of Youla–Kucera parameters is convex are those which are quadratically invariant. We further show that under additional assumptions, quadratic invariance is necessary and sufficient for the set of achievable Closed-loop maps to be convex. We give two versions of our results. The first applies to bounded linear operators on a Banach space and the second applies to (possibly unstable) causal LTI systems in discrete or continuous time.
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convexity of decentralized controller synthesis
arXiv: Systems and Control, 2013Co-Authors: Laurent Lessard, Sanjay LallAbstract:In decentralized control problems, a standard approach is to specify the set of allowable decentralized controllers as a Closed Subspace of linear operators. This then induces a corresponding set of Youla parameters. Previous work has shown that quadratic invariance of the controller set implies that the set of Youla parameters is convex. In this paper, we prove the converse. We thereby show that the only decentralized control problems for which the set of Youla parameters is convex are those which are quadratically invariant. We further show that under additional assumptions, quadratic invariance is necessary and sufficient for the set of achievable Closed-loop maps to be convex. We give two versions of our results. The first applies to bounded linear operators on a Banach space and the second applies to (possibly unstable) causal LTI systems in discrete or continuous time.
Heydar Radjavi - One of the best experts on this subject based on the ideXlab platform.
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on almost invariant Subspaces and approximate commutation
Journal of Functional Analysis, 2013Co-Authors: Laurent W Marcoux, Alexey I Popov, Heydar RadjaviAbstract:Abstract A Closed Subspace Y of a Banach space X is almost-invariant for a collection S of bounded linear operators on X if for each T ∈ S there exists a finite-dimensional Subspace F T of X such that T Y ⊆ Y + F T . In this paper, we study the existence of almost-invariant Subspaces for algebras of operators. We show, in particular, that if a Closed algebra of operators on a Hilbert space has a non-trivial almost-invariant Subspace then it has a non-trivial invariant Subspace. We also examine the structure of operators which admit a maximal commuting family of almost-invariant Subspaces. In particular, we prove that if T is an operator on a separable Hilbert space and if T P − P T has finite rank for all projections P in a given maximal abelian self-adjoint algebra M then T = M + F where M ∈ M and F is of finite rank.
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on almost invariant Subspaces and approximate commutation
arXiv: Functional Analysis, 2012Co-Authors: Laurent W Marcoux, Alexey I Popov, Heydar RadjaviAbstract:A Closed Subspace of a Banach space $\cX$ is almost-invariant for a collection $\cS$ of bounded linear operators on $\cX$ if for each $T \in \cS$ there exists a finite-dimensional Subspace $\cF_T$ of $\cX$ such that $T \cY \subseteq \cY + \cF_T$. In this paper, we study the existence of almost-invariant Subspaces of infinite dimension and codimension for various classes of Banach and Hilbert space operators. We also examine the structure of operators which admit a maximal commuting family of almost-invariant Subspaces. In particular, we prove that if $T$ is an operator on a separable Hilbert space and if $TP-PT$ has finite rank for all projections $P$ in a given maximal abelian self-adjoint algebra $\fM$ then $T=M+F$ where $M\in\fM$ and $F$ is of finite rank.
Contino Maximiliano - One of the best experts on this subject based on the ideXlab platform.
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SemiClosed projections and applications
'Elsevier BV', 2021Co-Authors: Contino Maximiliano, Maestripieri, Alejandra Laura, Marcantognini Palacios, Stefania Alma MaríaAbstract:We characterize the semiClosed projections and apply them to compute the Schur complement of a selfadjoint operator with respect to a Closed Subspace. These projections occur naturally when dealing with weak complementability.Fil: Contino, Maximiliano. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; ArgentinaFil: Maestripieri, Alejandra Laura. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Marcantognini Palacios, Stefania Alma María. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Área de Matemática
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SemiClosed projections and applications
2020Co-Authors: Contino Maximiliano, Maestripieri Alejandra, Marcantognini StefaniaAbstract:We characterize the semiClosed projections and apply them to compute the Schur complement of a selfadjoint operator with respect to a Closed Subspace. These projections occur naturally when dealing with weak complementability
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Schur complements of selfadjoint Krein space operators
2019Co-Authors: Contino Maximiliano, Maestripieri Alejandra, Marcantognini StefaniaAbstract:Given a bounded selfadjoint operator W on a Krein space H and a Closed Subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators
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Schur complements of selfadjoint Krein space operators
'Elsevier BV', 2019Co-Authors: Contino Maximiliano, Maestripieri, Alejandra Laura, Marcantognini Palacios, Stefania Alma MaríaAbstract:Given a bounded selfadjoint operator W on a Krein space H and a Closed Subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators.Fil: Contino, Maximiliano. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; ArgentinaFil: Maestripieri, Alejandra Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; ArgentinaFil: Marcantognini Palacios, Stefania Alma María. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
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Shorted operators and minus order
2018Co-Authors: Contino Maximiliano, Giribet, Juan Ignacio, Maestripieri AlejandraAbstract:Let $\mathcal{H}$ be a Hilbert space, $L(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$ and $W \in L(\mathcal{H})$ a positive operator. Given a Closed Subspace $\mathcal{S}$ of $\mathcal{H}$, we characterize the shorted operator $W_{/ \mathcal{S}}$ of $W$ to $\mathcal{S}$ as the maximum and as the infimum of certain sets, for the minus order $\stackrel{-}{\leq}.$ Also, given $A \in L(\mathcal{H})$ with Closed range, we study the following operator approximation problem considering the minus order: $$ min_{\stackrel{-}{\leq}} \ \{(AX-I)^*W(AX-I) : X \in L(\mathcal{H}), \mbox{ subject to } N(A^*W)\subseteq N(X) \}. $$ We show that, under certain conditions, the shorted operator $W_{/R(A)}$ (of $W$ to the range of $A$) is the minimum of this problem and we characterize the set of solutions