arXiv subjects

Masashi Wakaiki

Publications and source records attributed to Masashi Wakaiki.

At least 19 recordsLinked to original sources

arXiv2026

Operator-based data embedding for data-driven control of continuous-time systems from noisy data

We propose a data-driven method for designing state-feedback gains that achieve stabilization, $H_2$-control, and $H_\infty$-control for continuous-time systems. The state-input data are assumed to be corrupted by process noise, measurement noise, and input disturbances. We first characterize the set of all systems consistent with the noisy data using operator-based data embedding. This characterization yields necessary and sufficient conditions for data informativity under a certain class of noise. These conditions are formulated as linear matrix inequalities, and the feedback gains are constructed from their solutions. To enable direct controller design from noisy sampled data for continuous-time systems, we also obtain an upper bound on the reconstruction error of continuous-time signals.

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arXiv2026

Relation between semigroup growth and resolvent decay for immediately differentiable semigroups

We study the rate of growth of $\|AT(t)\|$ as $t \downarrow 0$ for an immediately differentiable $C_0$-semigroup $(T(t))_{t \geq 0}$ with generator $A$. We assume that the resolvent of the semigroup generator decays on the imaginary axis at rates described by functions of positive increase, which enable estimates on scales finer than polynomial ones. First, we present lower and upper bounds for the rates of growth of Banach space semigroups. Next, we improve the upper estimate for Hilbert space semigroups. Finally, for semigroups of normal operators on Hilbert spaces and multiplication $C_0$-semigroups on $L^p$-spaces, we establish an estimate that exactly captures the asymptotic behavior of $\|AT(t)\|$ as $t \downarrow 0$.

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arXiv2026

Data-driven control of continuous-time systems: A synthesis-operator approach

This paper addresses data-driven control of continuous-time systems. We develop a framework based on synthesis operators associated with state and input trajectories. A key advantage of the proposed method is that it does not require the state derivative and uses continuous-time data directly without sampling or filtering. First, systems consistent with the data are represented in terms of synthesis operators, into which the data trajectories are embedded. Next, we characterize data informativity properties for system identification and for stabilization in the noise-free case. Finally, we establish a necessary and sufficient condition for noisy data to be informative for quadratic stabilization. All these informativity characterizations are formulated in terms of finite-dimensional matrices, by leveraging the finite-rank structure of the synthesis operators.

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arXiv2026

Data-driven stabilization of continuous-time systems with noisy input-output data

We study data-driven stabilization of continuous-time systems in autoregressive form when only noisy input-output data are available. First, we provide an operator-based characterization of the set of systems consistent with the data. Next, combining this characterization with behavioral theory, we establish a necessary and sufficient condition for the noisy data to be informative for quadratic stabilization. This condition is formulated in terms of linear matrix inequalities, whose solutions yield a stabilizing controller. Finally, we characterize data informativity for system identification in the noise-free setting.

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arXiv2026

Characterization of decay rates for discrete operator semigroups

Let $T$ be a power-bounded linear operator on a Hilbert space $X$, and let $S$ be a bounded linear operator from another Hilbert space $Y$ to $X$. We investigate the non-exponential rate of decay of $\|T^nS\|$ as $n \to \infty$. First, when $X = Y$ and $S$ commutes with $T$, we characterize the decay rate of $\|T^nS\|$ in terms of the growth rate of $\|(λI - T)^{-k}S\|$ as $|λ| \downarrow 1$ for some $k \in \mathbb{N}$. Next, we provide another characterization by means of an integral estimate of $\|(λI - T)^{-k}S\|$. The second characterization is then applied to asymptotic estimates for perturbed discrete operator semigroups. Finally, we present some results on the relation between the decay rate of $\|T^nS\|$ and the boundedness of the sum $\sum_{n=1}^{\infty} f(n)\|T^nSy\|^p$ for all $y \in Y$ in the Banach space setting, where $f \colon \mathbb{N} \to(0,\infty)$ and $p \geq 1$.

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arXiv2025

Data informativity for stabilization of discrete-time infinite-dimensional systems

This paper develops a data-driven framework for stabilization of discrete-time infinite-dimensional systems. We investigate informativity for stabilization, defined as the existence of a feedback gain that stabilizes all systems compatible with the available input-state data. Assuming that infinite-length data are Bessel sequences, we first establish a sufficient condition for data informativity in the noise-free case. We next show that this sufficient condition is also necessary under a mild data assumption when the input space is one-dimensional. Furthermore, if the state sequence forms a frame, then the sufficient condition can be extended to the case of noisy data. Finally, when the unstable part of the true system is known to be finite-dimensional, we derive a necessary and sufficient condition for data informativity of finite-length data.

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arXiv2025

Stabilization of infinite-dimensional systems under quantization and packet loss

We study the problem of stabilizing infinite-dimensional systems with input and output quantization. The closed-loop system we consider is subject to packet loss, whose average duration is assumed to be bounded. Given a bound on the initial state, we propose a design method for dynamic quantizers with zoom parameters. We show that the closed-loop state starting in a given region exponentially converges to zero if bounds on quantization errors and packet-loss intervals satisfy suitable conditions. Since the norms of the operators representing the system dynamics are used in the proposed quantizer design, we also present methods for approximately computing the operator norms.

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arXiv2025

Self-triggered Stabilization of Contracting Systems under Quantization

We propose self-triggered control schemes for nonlinear systems with quantized state measurements. Our focus lies on scenarios where both the controller and the self-triggering mechanism receive only the quantized state at each sampling time. We assume that the ideal closed-loop system without quantization or self-triggered sampling is contracting. Moreover, an upper bound on the growth rate of the open-loop system is assumed to be known. We present two control schemes that achieve closed-loop stability without Zeno behavior. The first scheme is implemented under logarithmic quantization and uses the quantized state for the threshold in the triggering condition. The second one is a joint design of zooming quantization and self-triggered sampling, where the adjustable zoom parameter for quantization changes based on inter-sampling times and is also used for the threshold of self-triggered sampling. In both schemes, the self-triggering mechanism predicts the future state from the quantized data for the computation of the next sampling time. We employ a trajectory-based approach for stability analysis, where contraction theory plays a key role.

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arXiv2025

Characterizations of the Crandall--Pazy Class of $C_0$-semigroups on Hilbert Spaces and Their Application to Decay Estimates

We investigate immediately differentiable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$ satisfying $\sup_{0 < t <1} t^{1/β}\|Ae^{-tA}\| < \infty$ for some $0 < β\leq 1$. Such $C_0$-semigroups are referred to as the Crandall--Pazy class of $C_0$-semigroups. In the Hilbert space setting, we present two characterizations of the Crandall--Pazy class. We then apply these characterizations to estimate decay rates for Crank--Nicolson schemes with smooth initial data when the associated abstract Cauchy problem is governed by an exponentially stable $C_0$-semigroup in the Crandall--Pazy class. The first approach is based on a functional calculus called the $\mathcal{B}$-calculus. The second approach builds upon estimates derived from Lyapunov equations and improves the decay estimate obtained in the first approach, under the additional assumption that $-A^{-1}$ generates a bounded $C_0$-semigroup.

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arXiv2024

Decay estimates for Cayley transforms and inverses of semigroup generators via the $\mathcal{B}$-calculus

Let $-A$ be the generator of a bounded $C_0$-semigroup $(e^{-tA})_{t \geq 0}$ on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform $V_ω(A) := (A-ωI) (A+ωI)^{-1}$ with $ω>0$. We give a decay estimate for $\|V_ω(A)^nA^{-1}\|$ when $(e^{-tA})_{t \geq 0}$ is polynomially stable. Considering the case where the parameter $ω$ varies, we estimate $\|(\prod_{k=1}^n V_{ω_k}(A))A^{-1}\|$ for exponentially stable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$. Next we show that if the generator $-A$ of the bounded $C_0$-semigroup has a bounded inverse, then $\sup_{t \geq 0} \|e^{-tA^{-1}} A^{-α} \| < \infty$ for all $α>0$. We also present an estimate for the rate of decay of $\|e^{-tA^{-1}} A^{-1} \|$, assuming that $(e^{-tA})_{t \geq 0}$ is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the $\mathcal{B}$-calculus.

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arXiv2024

Decay of Operator Semigroups, Infinite-time Admissibility, and Related Resolvent Estimates

We study decay rates for bounded $C_0$-semigroups from the perspective of $L^p$-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the resolvent behavior in the open right half-plane. A similar characterization based on $L^p$-infinite-time admissibility is provided for multiplication semigroups on $L^q$-spaces with $1 \leq q \leq p < \infty$. For polynomially stable $C_0$-semigroups on Hilbert spaces, we also give a sufficient condition for $L^2$-infinite-time admissibility.

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arXiv2023

Decay Rate of $\exp(A^{-1}t)A^{-1}$ on a Hilbert Space and the Crank-Nicolson Scheme with Smooth Initial Data

This paper is concerned with the decay rate of $e^{A^{-1}t}A^{-1}$ for the generator $A$ of an exponentially stable $C_0$-semigroup on a Hilbert space. To estimate the decay rate of $e^{A^{-1}t}A^{-1}$, we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable $C_0$-semigroup whose generator is normal.

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arXiv2023

Self-triggered Consensus of Multi-agent Systems with Quantized Relative State Measurements

This paper addresses the consensus problem of first-order continuous-time multi-agent systems over undirected graphs. Each agent samples relative state measurements in a self-triggered fashion and transmits the sum of the measurements to its neighbors. Moreover, we use finite-level dynamic quantizers and apply the zooming-in technique. The proposed joint design method for quantization and self-triggered sampling achieves asymptotic consensus, and inter-event times are strictly positive. Sampling times are determined explicitly with iterative procedures including the computation of the Lambert $W$-function. A simulation example is provided to illustrate the effectiveness of the proposed method.

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arXiv2023

Robustness of Polynomial Stability with Respect to Sampling

We provide a partially affirmative answer to the following question on robustness of polynomial stability with respect to sampling: ``Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system transferred to the sampled-data system under sufficiently fast sampling?'' The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.

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arXiv2023

Self-triggered Resilient Stabilization of Linear Systems with Quantized Output

This paper studies the problem of stabilizing a self-triggered control system with quantized output. Employing a standard observer-based state feedback control law, a self-triggering mechanism that dictates the next sampling time based on quantized output is co-developed with an output encoding scheme. If, in addition, the transmission protocols at the controller-to-actuator (C-A) and sensor-to-controller (S-C) channels can be adapted, the self-triggered control architecture can be considerably simplified, leveraging a delicate observer-based deadbeat controller to eliminate the need for running the controller in parallel at the encoder side. To account for denial-of-service (DoS) in the S-C channel, the proposed output encoding and self-triggered control schemes are further made resilient. It is shown that a linear time-invariant system can be exponentially stabilized if some conditions on the average DoS duration time are met. There is a trade-off between the maximum inter-sampling time and the resilience against DoS attacks. Finally, a numerical example is presented to demonstrate the practical merits of the proposed self-triggered control schemes and associated theory.

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arXiv2022

Stability Analysis of Infinite-dimensional Event-triggered and Self-triggered Control Systems with Lipschitz Perturbations

This paper addresses the following question: "Suppose that a state-feedback controller stabilizes an infinite-dimensional linear continuous-time system. If we choose the parameters of an event/self-triggering mechanism appropriately, is the event/self-triggered control system stable under all sufficiently small nonlinear Lipschitz perturbations?" We assume that the stabilizing feedback operator is compact. This assumption is used to guarantee the strict positiveness of inter-event times and the existence of the mild solution of evolution equations with unbounded control operators. First, for the case where the control operator is bounded, we show that the answer to the above question is positive, giving a sufficient condition for exponential stability, which can be employed for the design of event/self-triggering mechanisms. Next, we investigate the case where the control operator is unbounded and prove that the answer is still positive for periodic event-triggering mechanisms.

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arXiv2022

Semi-uniform Input-to-state Stability of Infinite-dimensional Systems

We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on attractivity properties as in the uniform case. Sufficient conditions for linear systems to be polynomially input-to-state stable are provided, which restrict the range of the input operator depending on the rate of polynomial decay of the product of the semigroup and the resolvent of its generator. We also show that a class of bilinear systems are polynomially integral input-to-state stable under a certain smoothness assumption on nonlinear operators.

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arXiv2022

Self-triggered Stabilization of Discrete-time Linear Systems with Quantized State Measurements

We study the self-triggered stabilization of discrete-time linear systems with quantized state measurements. In the networked control system we consider, sensors may be spatially distributed and be connected to a self-triggering mechanism through finite data-rate channels. Each sensor independently encodes its measurements and sends them to the self-triggering mechanism. The self-triggering mechanism integrates quantized measurement data and then computes sampling times. Assuming that the closed-loop system is stable in the absence of quantization and self-triggered sampling, we propose a joint design method of an encoding scheme and a self-triggering mechanism for stabilization. To deal with data inaccuracy due to quantization, the proposed self-triggering mechanism uses not only quantized data but also an upper bound of quantization errors, which is shared with a decoder.

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