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Abstract: Abstract Classical turnpikes correspond to optimal steady states which are attractors of infinite-horizon optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions projected onto a symmetry-induced manifold coincide with those of a reduced-order problem defined on the manifold under certain conditions. We also propose sufficient conditions for the existence of manifold turnpikes based on a tailored notion of dissipativity with respect to manifolds. Furthermore, we show how the classical Legendre transformation between Euler–Lagrange and Hamilton formalisms can be extended to the adjoint variables. Finally, we draw upon the Kepler problem to illustrate our findings. PubDate: 2022-05-03

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Abstract: Abstract We study output reference tracking of systems with high relative degree via output feedback only; this is, tracking where the output derivatives are unknown. To this end, we prove that the conjunction of the funnel pre-compensator with a minimum phase system of arbitrary relative degree yields a system of the same relative degree which is minimum phase as well. The error between the original system’s output and the pre-compensator’s output evolves within a prescribed performance funnel; moreover, the derivatives of the funnel pre-compensator’s output are known explicitly. Therefore, output reference tracking with prescribed transient behaviour of the tracking error is possible without knowledge of the derivatives of the original system’s output; via funnel control schemes for instance. PubDate: 2022-04-27

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Abstract: Abstract We present a circle criterion which is necessary and sufficient for absolute stability with respect to a natural class of sector-bounded nonlinear causal operators. This generalized circle criterion contains the classical result as a special case. Furthermore, we develop a version of the generalized criterion which guarantees input-to-state stability. PubDate: 2022-04-18

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Abstract: Abstract An online policy learning problem of linear control systems is studied. In this problem, the control system is known and linear, and a sequence of quadratic cost functions is revealed to the controller in hindsight, and the controller updates its policy to achieve a sublinear regret, similar to online optimization. A modified online Riccati algorithm is introduced that under some boundedness assumption leads to logarithmic regret bound. In particular, the logarithmic regret for the scalar case is achieved without boundedness assumption. Our algorithm, while achieving a better regret bound, also has reduced complexity compared to earlier algorithms which rely on solving semi-definite programs at each stage. PubDate: 2022-04-07

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Abstract: Abstract We study the optimal portfolio selection problem for the class of strategies which do not use direct observations of the appreciation rates of the prices, but rather use historical prices. We consider a multi-stock incomplete diffusion market model with random coefficients. An explicit solution exploring a modification of certainty equivalence principle is found for case of power utilities and for a case when the problem can be embedded to a Markovian setting. Some new estimators for the appreciation rates are given. PubDate: 2022-03-19

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Abstract: Abstract This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of the second-order moments of the system variables. Small-gain-theorem bounds are obtained for the back-action of the observer on the covariance dynamics of the plant in terms of the plant–observer coupling. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant–observer coupling matrix in order to mitigate the back-action effect. For the discounted mean square optimal CQF problem with penalized back-action, we establish the first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic techniques, this set of equations is represented in a more explicit form for equally dimensioned plant and observer. For a class of such observers with autonomous estimation error dynamics, we obtain a solution of the CQF problem and outline a homotopy method. The computation of the performance criteria and the observer synthesis are illustrated by numerical examples. PubDate: 2022-03-05 DOI: 10.1007/s00498-022-00315-4

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Abstract: Abstract In this paper, we establish the exponential BV stability of general systems of discretized scalar conservation laws with positive speed. The focus is on numerical approximation of such systems using a wide class of slope limiter schemes built from the upwind monotone flux. The proof is based on a Lyapunov analysis taken from the continuous theory (Coron et al. in J Differ Equ 262(1):1–30, 2017) and a careful use of Harten formalism. PubDate: 2022-03-01 DOI: 10.1007/s00498-021-00301-2

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Abstract: Abstract In this article, the robust Stackelberg controllability (RSC) problem is studied for a nonlinear fourth-order parabolic equation, namely the Kuramoto–Sivashinsky equation. When three external sources are acting into the system, the RSC problem consists essentially in combining two subproblems: the first one is a saddle point problem among two sources. Such sources are called the “follower control” and its associated “disturbance signal.” This procedure corresponds to a robust control problem. The second one is a hierarchic control problem (Stackelberg strategy), which involves the third force, so-called leader control. The RSC problem establishes a simultaneous game for these forces in the sense that the leader control has as objective to verify a controllability property, while the follower control and perturbation solve a robust control problem. In this paper, the leader control obeys to the exact controllability to the trajectories. Additionally, iterative algorithms to approximate the robust control problem as well as the robust Stackelberg strategy for the nonlinear Kuramoto–Sivashinsky equation are developed and implemented. PubDate: 2022-02-24 DOI: 10.1007/s00498-022-00316-3

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Abstract: Abstract In this work, we deal with the exponential stability of the nonlinear Korteweg–de Vries equation on a finite star-shaped network in the presence of delayed internal feedback. We start by proving the well-posedness of the system and some regularity results. Then, we state an exponential stabilization result using a Lyapunov function by imposing small initial data and a restriction over the lengths. In this part also, we are able to obtain an explicit expression for the decay rate. Then, we prove a semi-global exponential stability result, which is based on an observability inequality working directly on the nonlinear system. Next, we study the case where it may happen that a control domain with delay is outside the control domain without delay. In that case, we obtain also a local exponential stabilization result. Finally, we present some numerical simulations to illustrate the stabilization. PubDate: 2022-02-24 DOI: 10.1007/s00498-022-00319-0

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Abstract: Abstract In this paper, the potential differential game concept introduced by Fonseca-Morales and Hernández-Lerma (2018) is used in analyzing stabilization problems for n-player noncooperative capital accumulation games (CAGs). By first identifying a CAG as a potential game, an associated optimal control problem (OCP) of the CAG is obtained, whose optimal solution is an open-loop Nash equilibrium for the CAG. Compared with a saddle-point stability condition obtained for undiscounted CAG in the literature, a sufficient and easily verifiable condition is obtained for both discounted and undiscounted CAGs. In addition, the concept allows the turnpike property obtained for OCPs in Trélat and Zuazua (2015) to be verified for CAGs. Lastly, an illustrative example is given to verify the latter stability result for some CAGs. PubDate: 2022-02-19 DOI: 10.1007/s00498-022-00314-5

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Abstract: Abstract A discrete multidimensional system is the set of solutions to a system of linear partial difference equations defined on the lattice \(\mathbb {Z}^n\) . This paper shows that it is determined by a unique coarsest sublattice, in the sense that the solutions of the system on this sublattice determine the solutions on \(\mathbb {Z}^n\) ; it is therefore the correct domain of definition of the discrete system. In turn, the defining sublattice is determined by a Galois group of symmetries that leave invariant the equations defining the system. These results find application in understanding properties of the system such as controllability and autonomy, and in its order reduction. PubDate: 2022-02-16 DOI: 10.1007/s00498-022-00318-1

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Abstract: Abstract Let \({\mathrm {Sl}}\left( n,{\mathbb {H}}\right) \) be the Lie group of \(n\times n \) quaternionic matrices g with \(\left \det g\right =1\) . We prove that a subsemigroup \(S \subset {\mathrm {Sl}}\left( n,{\mathbb {H}}\right) \) with nonempty interior is equal to \({\mathrm {Sl}}\left( n,{\mathbb {H}}\right) \) if S contains a special subgroup isomorphic to \({\mathrm {Sl}}\left( 2,{\mathbb {H}}\right) \) . From this, we give sufficient conditions on \(A,B\in \mathfrak {sl}\left( n, {\mathbb {H}}\right) \) to ensure that the invariant control system \({\dot{g}} =Ag+uBg\) is controllable on \({\mathrm {Sl}}\left( n,{\mathbb {H}}\right) \) . We prove also that these conditions are generic in the sense that we obtain an open and dense set of controllable pairs \(\left( A,B\right) \in \mathfrak {sl}\left( n,{\mathbb {H}}\right) ^{2}\) . PubDate: 2022-02-11 DOI: 10.1007/s00498-022-00317-2

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Abstract: Abstract We study integral input-to-state stability of bilinear systems with unbounded control operators and derive natural sufficient conditions. The results are applied to a bilinearly controlled Fokker–Planck equation. PubDate: 2022-02-07 DOI: 10.1007/s00498-021-00308-9

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Abstract: Abstract We present a Reinforcement Learning (RL) algorithm to solve infinite horizon asymptotic Mean Field Game (MFG) and Mean Field Control (MFC) problems. Our approach can be described as a unified two-timescale Mean Field Q-learning: The same algorithm can learn either the MFG or the MFC solution by simply tuning the ratio of two learning parameters. The algorithm is in discrete time and space where the agent not only provides an action to the environment but also a distribution of the state in order to take into account the mean field feature of the problem. Importantly, we assume that the agent cannot observe the population’s distribution and needs to estimate it in a model-free manner. The asymptotic MFG and MFC problems are also presented in continuous time and space, and compared with classical (non-asymptotic or stationary) MFG and MFC problems. They lead to explicit solutions in the linear-quadratic (LQ) case that are used as benchmarks for the results of our algorithm. PubDate: 2022-01-15 DOI: 10.1007/s00498-021-00310-1

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Abstract: Abstract For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded. PubDate: 2021-11-12 DOI: 10.1007/s00498-021-00311-0

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Abstract: Abstract We propose a neural network approach to model general interaction dynamics and an adjoint-based stochastic gradient descent algorithm to calibrate its parameters. The parameter calibration problem is considered as optimal control problem that is investigated from a theoretical and numerical point of view. We prove the existence of optimal controls, derive the corresponding first-order optimality system and formulate a stochastic gradient descent algorithm to identify parameters for given data sets. To validate the approach, we use real data sets from traffic and crowd dynamics to fit the parameters. The results are compared to forces corresponding to well-known interaction models such as the Lighthill–Whitham–Richards model for traffic and the social force model for crowd motion. PubDate: 2021-10-27 DOI: 10.1007/s00498-021-00309-8

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Abstract: Abstract In linear system theory, it is a well-known fact that a regulator given by the cascade of an oscillatory dynamics, driven by some regulated variables, and of a stabiliser stabilising the cascade of the plant and of the oscillators has the ability of blocking on the steady state of the regulated variables any harmonics matched with the ones of the oscillators. This is the well-celebrated internal model principle. In this paper, we are interested to follow the same design route for a controlled plant that is a nonlinear and periodic system with period T: we add a bunch of linear oscillators, embedding \(n_o\) harmonics that are multiple of \(2 \pi /T\) , driven by a “regulated variable” of the nonlinear system, we look for a stabiliser for the nonlinear cascade of the plant and the oscillators, and we study the asymptotic properties of the resulting closed-loop regulated variable. In this framework, the contributions of the paper are multiple: for specific class of minimum-phase systems we present a systematic way of designing a stabiliser, which is uniform with respect to \(n_o\) , by using a mix of high-gain and forwarding techniques; we prove that the resulting closed-loop system has a periodic steady state with period T with a domain of attraction not shrinking with \(n_o\) ; similarly to the linear case, we also show that the spectrum of the steady-state closed-loop regulated variable does not contain the n harmonics embedded in the bunch of oscillators and that the \(L_2\) norm of the regulated variable is a monotonically decreasing function of \(n_o\) . The results are robust, namely the asymptotic properties on the regulated variable hold also in the presence of any uncertainties in the controlled plant not destroying closed-loop stability. PubDate: 2021-10-26 DOI: 10.1007/s00498-021-00307-w

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Abstract: Abstract In this work, we consider impulsive dynamical systems evolving on an infinite-dimensional space and subjected to external perturbations. We look for stability conditions that guarantee the input-to-state stability for such systems. Our new dwell-time conditions allow the situation, where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov like methods are developed for this purpose. Illustrative finite and infinite dimensional examples are provided to demonstrate the application of the main results. These examples cannot be treated by any other published approach and demonstrate the effectiveness of our results. PubDate: 2021-09-22 DOI: 10.1007/s00498-021-00305-y

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Abstract: Abstract This paper develops the analysis of discrete-time periodically time-varying linear systems over finite fields. It is shown that the conditions for the existence of Floquet Transform for periodic linear systems over reals (or complex) do not carry forward for this case over finite fields. The existence of Floquet Transform is shown to be equivalent to the existence of an Nth root of the monodromy matrix for the class of non-singular periodic linear systems. As the verification of existence and computation of the Nth root is a computationally hard problem, an independent analysis of the solutions of such systems is carried without the use of Floquet Transform. It is proved that all initial conditions of such systems lie either in a periodic orbit or a chain leading to a periodic orbit. A subspace of the state space is also identified, containing all initial conditions lying on a periodic orbit. Further, whenever the Floquet Transform exists, more concrete results on the orbit lengths are established depending on the coprimeness of the orbit length with the system period. PubDate: 2021-09-07 DOI: 10.1007/s00498-021-00304-z