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Tomonari Suzuki - One of the best experts on this subject based on the ideXlab platform.
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the set of common fixed points of a one parameter Continuous Semigroup of nonexpansive mappings is f frac 1 2 t 1 frac 1 2 t sqrt 2 in strictly convex banach spaces
Taiwanese Journal of Mathematics, 2006Co-Authors: Tomonari SuzukiAbstract:In this paper, we prove the following. Let $E$ be a strictly convex Banach space. Let $\{ T(t) : t \geq 0 \}$ be a one-parameter strongly Continuous Semigroup of nonexpansive mappings on a subset $C$ of $E$. Then \[ \bigcap_{t \geq 0} F(T(t)) = F\left( \frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}) \right) \] holds, where $F(T(t))$ is the set of fixed points of $T(t)$ for each $t \geq 0$.
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the set of common fixed points of a one parameter Continuous Semigroup of nonexpansive mappings is f in strictly convex banach spaces
Taiwanese Journal of Mathematics, 2006Co-Authors: Tomonari SuzukiAbstract:In this paper, we prove the following. Let E be a strictly convex Banach space. Let {T(t) : t ≥ 0} be a one-parameter strongly Continuous Semigroup of nonexpansive mappings on a subset C of E. Then t≥0 F T(t) = F 1 2T(1) + 1 2T( √ 2) holds, where F T(t) is the set of fixed points of T(t) for each t ≥ 0.
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the set of common fixed points of a one parameter Continuous Semigroup of mappings is f t 1 f t 2
Proceedings of the American Mathematical Society, 2006Co-Authors: Tomonari SuzukiAbstract:In this paper we prove the following theorem: Let {T(t): t ≥ 0} be a one-parameter Continuous Semigroup of mappings on a subset C of a Banach space E. The set of all fixed points of T(t) is denoted by F(T(t)) for each t > 0. Then ∩ F(T(t)) = F(T(1)) n F(T(√2)) t>0 holds. Using this theorem, we discuss convergence theorems to a common fixed point of {T(t): t > 0}.
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the set of common fixed points of an n parameter Continuous Semigroup of mappings
Nonlinear Analysis-theory Methods & Applications, 2005Co-Authors: Tomonari SuzukiAbstract:Abstract In this paper, using Kronecker's theorem, we discuss the set of common fixed points of an n-parameter Continuous Semigroup { T ( p ) : p ∈ R + n } of mappings. We also discuss some convergence theorems to a common fixed point of an n-parameter nonexpansive Semigroup { T ( p ) : p ∈ R + n } without using the Bochner integral.
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strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive Semigroups without bochner integrals
Journal of Mathematical Analysis and Applications, 2005Co-Authors: Tomonari SuzukiAbstract:Abstract In this paper, we prove Krasnoselskii and Mann's type convergence theorems for nonexpansive Semigroups without using Bochner integral and without assuming the strict convexity of Banach spaces. One of our main results is the following: let C be a compact convex subset of a Banach space E and let { T ( t ) : t ⩾ 0 } be a one-parameter strongly Continuous Semigroup of nonexpansive mappings on C. Let { t n } be a sequence in [ 0 , ∞ ) satisfying lim inf n → ∞ t n lim sup n → ∞ t n and lim n → ∞ ( t n + 1 − t n ) = 0 . Let λ ∈ ( 0 , 1 ) . Define a sequence { x n } in C by x 1 ∈ C and x n + 1 = λ T ( t n ) x n + ( 1 − λ ) x n for n ∈ N . Then { x n } converges strongly to a common fixed point of { T ( t ) : t ⩾ 0 } .
Susan A Frost - One of the best experts on this subject based on the ideXlab platform.
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direct adaptive control of non minimum phase linear infinite dimensional systems in hilbert space using a zero dynamics estimator
Conference on Decision and Control, 2019Co-Authors: Mark J Balas, Susan A FrostAbstract:Linear infinite dimensional systems are described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on a general Hilbert space of states and are controlled via a finite number of actuators and sensors. Many distributed applications are included in this formulation, such as large flexible aerospace structures, adaptive optics, diffusion reactions, smart electric power grids, and quantum information systems. Using a recently developed normal form for these systems, we have developed the following stability result: an infinite dimensional linear system is Almost Strictly Dissipative (ASD) if and only if its high frequency gain CB is symmetric and positive definite and the open loop system is minimum phase, i.e. its transmission zeros are all exponentially stable.In this paper, we focus on infinite dimensional linear systems that are non-minimum phase because a finite number of zeros are unstable. We previously developed a blending method to compensate for this issue where we modify or "blend" the output of the infinite dimensional plant, and then control this modified output rather than the original control output. In this paper we use a finite dimensional zero dynamics estimator based on a modified output but use the estimator to produce a fully minimum phase system. Then direct adaptive control for the infinite dimensional plant can focus on the original control output rather than the modified output. These results are illustrated by application to direct adaptive control of general linear systems on a Hilbert space that are described by self-adjoint operators with compact resolvent.
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sensor blending for direct adaptive control of non minimum phase linear infinite dimensional systems in hilbert space
Advances in Computing and Communications, 2017Co-Authors: Mark J Balas, Susan A FrostAbstract:Linear infinite dimensional systems are described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on a general Hilbert space of states and are controlled via a finite number of actuators and sensors. Many distributed applications are included in this formulation, such as large flexible aerospace structures, adaptive optics, diffusion reactions, smart electric power grids, and quantum information systems.
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adaptive model tracking control for weakly minimum phase linear infinite dimensional systems in hilbert space using a zero filter
AIAA Guidance Navigation and Control Conference, 2016Co-Authors: Mark J Balas, Susan A FrostAbstract:Abstract: Given a linear Continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing direct model reference adaptive control law with persistent disturbance rejection and robustness properties. The plant is described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on the Hilbert space of states. For this paper, the plant will be weakly minimum phase, i.e. there will be a finite number of unstable zeros with real part equal to zero. All other zeros will be exponentially stable. The central result will show that all errors will converge to a prescribed neighborhood of zero in an infinite dimensional Hilbert space even though the plant is not truly minimum phase. The result will not require the use of the standard Barbalat Lemma which requires certain signals to be uniformly Continuous. This result is used to determine conditions under which a linear Infinite-dimensional system can be directly adaptively controlled to follow a reference model. In particular we examine conditions for a set of ideal trajectories to exist for the tracking problem. Our principal result will be that the direct adaptive controller can be compensated with a zero filter for the unstable zeros which will produce the desired robust adaptive control results even though the plant is only weakly minimum phase. Our results are applied to adaptive control of general linear infinite dimensional systems described by self-adjoint operators with compact resolvent.
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adaptive control for weakly minimum phase linear infinite dimensional systems in hilbert space using a zero filter
2016Co-Authors: Mark J Balas, Susan A FrostAbstract:Given a linear Continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing direct model reference adaptive control law with persistent disturbance rejection and robustness properties. The plant is described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on the Hilbert space of states. For this paper, the plant will be weakly minimum phase, i.e., there will be a finite number of unstable zeros with real part equal to zero. All other zeros will be exponentially stable.
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adaptive regulation in the presence of persistent disturbances for linear infinite dimensional systems in hilbert space conditions for almost strict dissipativity
European Control Conference, 2015Co-Authors: Mark J Balas, Susan A FrostAbstract:This paper is focused on adaptively controlling a linear infinite-dimensional system to cause it to regulate the output to zero in the presence of persistent disturbances. The plant (A, B, C) is described by a closed, densely defined linear operator A that generates a Continuous Semigroup of bounded operators on a Hilbert space of states; the input-output operators B & C are finite rank linear operators. We show that there exists a direct model reference adaptive control law that regulates the output in the presence of disturbances of known waveform but unknown amplitude and phase. The conditions needed for the success of the direct adaptive controller include the need for (A, B, C) to be almost strictly dissipative (ASD). In finite dimensional space, ASD is equivalent to two simple open-loop requirements: the high frequency gain CB is sign-definite and the open-loop transfer function P(s) is minimum phase. Our main result will prove infinite-dimensional versions of these conditions for a large class of infinite-dimensional systems.
Mark J Balas - One of the best experts on this subject based on the ideXlab platform.
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direct adaptive control of non minimum phase linear infinite dimensional systems in hilbert space using a zero dynamics estimator
Conference on Decision and Control, 2019Co-Authors: Mark J Balas, Susan A FrostAbstract:Linear infinite dimensional systems are described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on a general Hilbert space of states and are controlled via a finite number of actuators and sensors. Many distributed applications are included in this formulation, such as large flexible aerospace structures, adaptive optics, diffusion reactions, smart electric power grids, and quantum information systems. Using a recently developed normal form for these systems, we have developed the following stability result: an infinite dimensional linear system is Almost Strictly Dissipative (ASD) if and only if its high frequency gain CB is symmetric and positive definite and the open loop system is minimum phase, i.e. its transmission zeros are all exponentially stable.In this paper, we focus on infinite dimensional linear systems that are non-minimum phase because a finite number of zeros are unstable. We previously developed a blending method to compensate for this issue where we modify or "blend" the output of the infinite dimensional plant, and then control this modified output rather than the original control output. In this paper we use a finite dimensional zero dynamics estimator based on a modified output but use the estimator to produce a fully minimum phase system. Then direct adaptive control for the infinite dimensional plant can focus on the original control output rather than the modified output. These results are illustrated by application to direct adaptive control of general linear systems on a Hilbert space that are described by self-adjoint operators with compact resolvent.
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sensor blending for direct adaptive control of non minimum phase linear infinite dimensional systems in hilbert space
Advances in Computing and Communications, 2017Co-Authors: Mark J Balas, Susan A FrostAbstract:Linear infinite dimensional systems are described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on a general Hilbert space of states and are controlled via a finite number of actuators and sensors. Many distributed applications are included in this formulation, such as large flexible aerospace structures, adaptive optics, diffusion reactions, smart electric power grids, and quantum information systems.
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adaptive model tracking control for weakly minimum phase linear infinite dimensional systems in hilbert space using a zero filter
AIAA Guidance Navigation and Control Conference, 2016Co-Authors: Mark J Balas, Susan A FrostAbstract:Abstract: Given a linear Continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing direct model reference adaptive control law with persistent disturbance rejection and robustness properties. The plant is described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on the Hilbert space of states. For this paper, the plant will be weakly minimum phase, i.e. there will be a finite number of unstable zeros with real part equal to zero. All other zeros will be exponentially stable. The central result will show that all errors will converge to a prescribed neighborhood of zero in an infinite dimensional Hilbert space even though the plant is not truly minimum phase. The result will not require the use of the standard Barbalat Lemma which requires certain signals to be uniformly Continuous. This result is used to determine conditions under which a linear Infinite-dimensional system can be directly adaptively controlled to follow a reference model. In particular we examine conditions for a set of ideal trajectories to exist for the tracking problem. Our principal result will be that the direct adaptive controller can be compensated with a zero filter for the unstable zeros which will produce the desired robust adaptive control results even though the plant is only weakly minimum phase. Our results are applied to adaptive control of general linear infinite dimensional systems described by self-adjoint operators with compact resolvent.
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adaptive control for weakly minimum phase linear infinite dimensional systems in hilbert space using a zero filter
2016Co-Authors: Mark J Balas, Susan A FrostAbstract:Given a linear Continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing direct model reference adaptive control law with persistent disturbance rejection and robustness properties. The plant is described by a closed, densely defined linear operator that generates a Continuous Semigroup of bounded operators on the Hilbert space of states. For this paper, the plant will be weakly minimum phase, i.e., there will be a finite number of unstable zeros with real part equal to zero. All other zeros will be exponentially stable.
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adaptive regulation in the presence of persistent disturbances for linear infinite dimensional systems in hilbert space conditions for almost strict dissipativity
European Control Conference, 2015Co-Authors: Mark J Balas, Susan A FrostAbstract:This paper is focused on adaptively controlling a linear infinite-dimensional system to cause it to regulate the output to zero in the presence of persistent disturbances. The plant (A, B, C) is described by a closed, densely defined linear operator A that generates a Continuous Semigroup of bounded operators on a Hilbert space of states; the input-output operators B & C are finite rank linear operators. We show that there exists a direct model reference adaptive control law that regulates the output in the presence of disturbances of known waveform but unknown amplitude and phase. The conditions needed for the success of the direct adaptive controller include the need for (A, B, C) to be almost strictly dissipative (ASD). In finite dimensional space, ASD is equivalent to two simple open-loop requirements: the high frequency gain CB is sign-definite and the open-loop transfer function P(s) is minimum phase. Our main result will prove infinite-dimensional versions of these conditions for a large class of infinite-dimensional systems.
Jan Poland - One of the best experts on this subject based on the ideXlab platform.
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the critical spectrum of a strongly Continuous Semigroup
Advances in Mathematics, 2000Co-Authors: Rainer Nagel, Jan PolandAbstract:For a strongly Continuous Semigroup (T(t))t⩾0 with generator A we introduce its critical spectrum σcrit(T(t)). This yields in an optimal way the spectral mapping theorem σ(T(t))=etσ(A)∪σcrit(T(t)) and improves classical stability results.
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on the spectral mapping theorem for perturbed strongly Continuous Semigroups
Archiv der Mathematik, 2000Co-Authors: Simon Brendle, Rainer Nagel, Jan PolandAbstract:We consider a strongly Continuous Semigroup \((T(t))_{t \geqq 0}\) with generator A on a Banach space X, an A-bounded perturbation B, and the Semigroup \((S(t))_{t \geqq 0}\) generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed Semigroup \((S(t))_{t \geqq 0}\) .
Rainer Nagel - One of the best experts on this subject based on the ideXlab platform.
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the critical spectrum of a strongly Continuous Semigroup
Advances in Mathematics, 2000Co-Authors: Rainer Nagel, Jan PolandAbstract:For a strongly Continuous Semigroup (T(t))t⩾0 with generator A we introduce its critical spectrum σcrit(T(t)). This yields in an optimal way the spectral mapping theorem σ(T(t))=etσ(A)∪σcrit(T(t)) and improves classical stability results.
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on the spectral mapping theorem for perturbed strongly Continuous Semigroups
Archiv der Mathematik, 2000Co-Authors: Simon Brendle, Rainer Nagel, Jan PolandAbstract:We consider a strongly Continuous Semigroup \((T(t))_{t \geqq 0}\) with generator A on a Banach space X, an A-bounded perturbation B, and the Semigroup \((S(t))_{t \geqq 0}\) generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed Semigroup \((S(t))_{t \geqq 0}\) .