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Abstract: In this paper, we will deal with a linear quadratic optimal control problem with unknown dynamics. As a modeling assumption, we will suppose that the knowledge that an agent has on the current system is represented by a probability distribution \(\pi \) on the space of matrices. Furthermore, we will assume that such a probability measure is opportunely updated to take into account the increased experience that the agent obtains while exploring the environment, approximating with increasing accuracy the underlying dynamics. Under these assumptions, we will show that the optimal control obtained by solving the “average” linear quadratic optimal control problem with respect to a certain \(\pi \) converges to the optimal control driven related to the linear quadratic optimal control problem governed by the actual, underlying dynamics. This approach is closely related to model-based reinforcement learning algorithms where prior and posterior probability distributions describing the knowledge on the uncertain system are recursively updated. In the last section, we will show a numerical test that confirms the theoretical results. PubDate: 2021-07-08

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Abstract: We study control properties of a linearized fluid–structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that as in the case without structure, the system is not exactly null-controllable in finite time. PubDate: 2021-07-07

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Abstract: In this paper we study a class of semilinear wave-type equations with viscoelastic damping and delay feedback with time variable coefficient. By combining semigroup arguments, careful energy estimates and an iterative approach we are able to prove, under suitable assumptions, a well-posedness result and an exponential decay estimate for solutions corresponding to small initial data. This extends and concludes the analysis initiated in Nicaise and Pignotti (J Evol Equ 15:107–129, 2015) and then developed in Komornik and Pignotti (Math Nachr, to appear, 2018), Nicaise and Pignotti (Evol Equ 18:947–971, 2018). PubDate: 2021-06-17 DOI: 10.1007/s00498-021-00292-0

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Abstract: This paper investigates the question of strong stabilizability of non-dissipative linear systems in Hilbert spaces with input saturation. It is proved under some verifiable conditions that the origin is asymptotically stable for the closed-loop semilinear systems. The contribution is then applied to the Schrödinger equation. PubDate: 2021-06-10 DOI: 10.1007/s00498-021-00291-1

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Abstract: Under a regularity assumption we prove that reachability in fixed time for nonlinear control systems is robust under control sampling. PubDate: 2021-06-04 DOI: 10.1007/s00498-021-00290-2

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Abstract: It is well known that there is a correspondence between convolutional codes and discrete-time linear systems over finite fields. In this paper, we employ the linear systems representation of a convolutional code to develop a decoding algorithm for convolutional codes over the erasure channel. In this kind of channel, which is important due to its use for data transmission over the Internet, the receiver knows if a received symbol is correct. We study the decoding problem using the state space description of a convolutional code, and this provides in a natural way additional information. With respect to previously known decoding algorithms, our new algorithm has the advantage that it is able to reduce the decoding delay as well as the computational effort in the erasure recovery process. We describe which properties a convolutional code should have in order to obtain a good decoding performance and illustrate it with an example. PubDate: 2021-06-04 DOI: 10.1007/s00498-021-00289-9

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Abstract: We extend the approach based on the linearization of triangular systems to new classes of non-linearizable control systems that are almost linearizable. This means that there exists a change of variables and control mapping all but one equations of the initial nonlinear system to a linear system. We show how this property can be used for solving the problem of constructive controllability, i.e., finding trajectories connecting two given points. Namely, we explicitly find a change of variables and control that maps \(n-1\) equations of the initial system to a linear system. For the remaining first-order nonlinear differential equation, which contains one unknown scalar parameter, the boundary value problem is considered. Once this one-dimensional problem is solved, a trajectory connecting two given points for the initial system is explicitly found. Moreover, we solve the stabilization problem for systems from the proposed classes under additional natural conditions. We give several examples to illustrate a constructive character of our approach. PubDate: 2021-05-15 DOI: 10.1007/s00498-021-00288-w

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Abstract: We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. https://doi.org/10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats. PubDate: 2021-05-15 DOI: 10.1007/s00498-021-00287-x

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Abstract: The main concern of this article is to investigate the boundary controllability of some \(2\times 2\) one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components. PubDate: 2021-05-05 DOI: 10.1007/s00498-021-00285-z

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Abstract: This paper considers dynamical systems over finite fields (DSFF) defined by a map in a vector space over a finite field. An associated linear dynamical system is constructed over the space of functions. This system constitutes the well known Koopman linear system framework of dynamical systems, hence called the Koopman linear system (KLS). It is first shown that several structural properties of solutions of the DSFF can be inferred from the solutions of the KLS. The KLS is then reduced to the smallest order (called RO-KLS) while still retaining all the information of the parameters of structure of solutions of the DSFF. Hence, the above computational problems of nonlinear DSFF are solvable by linear algebraic methods. It is also shown how fixed points, periodic points and roots of chains of the DSFF can be computed using the RO-KLS. Further, for DSFF with outputs, the output trajectories of the DSFF are in \(1-1\) correspondence with special class of output trajectories of RO-KLS and it is shown that the problem of nonlinear observability can be solved by a linear observer design for the RO-KLS. PubDate: 2021-04-24 DOI: 10.1007/s00498-021-00286-y

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Abstract: Positive dynamical or control systems have all their variables nonnegative. Euler discretization transforms a continuous-time system into a system on a discrete time scale. Some structural properties of the system may be preserved by discretization, while other may be lost. Four fundamental properties of positive systems are studied in the context of discretization: positivity, positive stability, positive reachability and positive observability. Both linear and nonlinear systems are investigated. PubDate: 2021-03-27 DOI: 10.1007/s00498-021-00283-1

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Abstract: In this paper, a dynamic regressor extension and mixed estimator is proposed with finite-time convergence and freedom to choose its time-varying adaptation gain and its derivation order. This freedom is exploited to enhance the transient and robustness performance of the estimation by analytically establishing the effects of both variables. The proposed estimator is used to design adaptive controllers and observers for nonlinear systems, which exhibit exponential order of convergence at an arbitrary rate of decay with robust and improved transient properties. These results are illustrated in a tracking control of nonlinear systems with parametric uncertainty. PubDate: 2021-03-22 DOI: 10.1007/s00498-021-00282-2

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Abstract: Poset-causal systems form a class of decentralized systems introduced by Shah and Parrilo (47th IEEE conference on decision and control, IEEE, 2008) and studied mainly in the context of optimal decentralized control. In this paper, we develop part of the realization theory for poset-causal systems. More specifically, we investigate several notions of controllability and observability, and their relation under duality. These new notions extend concepts of controllability and observability in the context of coordinated linear systems (Kempker et al. in Linear Algebra Appl 437:121–167, 2012). While for coordinated linear systems there is a clear hierarchical structure with a single (main) coordinator, for poset-causal systems there need not be a single coordinator, and the communication structure between the decentralized systems allows for more intricate structures, governed by partial orders. On the other hand, we show that the class of poset-causal systems is closed under duality, which is not the case for coordinated linear systems, and that duality relations between the various notions of observability and controllability exist. PubDate: 2021-03-20 DOI: 10.1007/s00498-021-00284-0

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Abstract: In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval \([0,\, T]\) and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals \([0, \, T]\) give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ is of the order \(1/\min \{t^n, (T-t)^n\}\) . We also show that a similar result holds for \(\epsilon \) -optimal solutions of the optimal control problems if \(\epsilon >0\) is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied. PubDate: 2021-03-10 DOI: 10.1007/s00498-021-00280-4

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Abstract: This paper is devoted to a studying of the controllability properties for the Coleman–Gurtin-type equation, which is a class of multidimensional integral–differential equations. The goal is to prove the existence of a control function which steers the state variable and the integral term to the neighborhood of two given final configurations at the same time, respectively. This new approximate controllability is defined by imposing some additional integral-type constraints on the usual approximate controllability, ensuring that the whole process reaches the neighborhood of the equilibrium. We also provide a characterization of the initial values, which can be driven to zero by a distributed control. The later is a supplement of non-null controllability for the Coleman–Gurtin model in the square integrable space. PubDate: 2021-03-05 DOI: 10.1007/s00498-021-00281-3

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Abstract: This manuscript presents a notion of parallelizability of control systems. Parallelizability is a well-known concept of dynamical systems that associates with complete instability and dispersiveness. The concept of dispersiveness has been successfully interpreted in the setup of control systems. This naturally asks about the meaning of a parallelizable control system. The answer can be given in the setting of control affine systems by evoking their control flows. The main result shows that a parallelizable control flow characterizes a dispersive control affine system. The dispersiveness is then equivalent to the existence of a functional with infinite limit at infinity. The results of the paper contribute to the controllability studies, since dispersive control systems admit no control set. For invariant control systems with commutative vector fields, null trace representative matrices are a necessary condition for the existence of control set. PubDate: 2021-02-22 DOI: 10.1007/s00498-021-00279-x

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Abstract: The following corrections should be made to this article. PubDate: 2021-02-05 DOI: 10.1007/s00498-021-00278-y

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Abstract: We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov functions for hybrid dynamical systems. Subject to some regularity of the dynamics, we show that Lyapunov inequalities can be checked only on a dense set and thus we avoid checking them at points of nondifferentiability of the Lyapunov function. Connections to other classes of locally Lipschitz or piecewise regular functions are also discussed, and applications to hybrid dynamical systems are included. PubDate: 2021-02-04 DOI: 10.1007/s00498-020-00273-9

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Abstract: Boundary feedback control design for systems of linear hyperbolic conservation laws in the presence of boundary measurements affected by disturbances is studied. The design of the controller is performed to achieve input-to-state stability (ISS) with respect to measurement disturbances with a minimal gain. The closed-loop system is analyzed as an abstract dynamical system with inputs. Sufficient conditions in the form of dissipation functional inequalities are given to establish an ISS bound for the closed-loop system. The control design problem is turned into an optimization problem over matrix inequality constraints. Semidefinite programming techniques are adopted to devise systematic control design algorithms reducing the effect of measurement disturbances. The effectiveness of the approach is extensively shown in several numerical examples. PubDate: 2021-01-02 DOI: 10.1007/s00498-020-00269-5