On relative controllability of delayed difference equations with multiple control functions

Pospíšil, Michal; Diblı́k, Josef; Fečkan, Michal

doi:10.1063/1.4912420

Cited by 10 publications

(10 citation statements)

References 13 publications

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“…Concerning the controllability problem, due to the infinite-dimensional nature of the dynamics of neutral functional differential equations and difference equations, several different notions of controllability can be used, such as exact, approximate, spectral, or relative controllability [5,30]. Relative controllability has been originally introduced in the study of control systems with delays in the control input [5,20,27], but this notion has later been extended and used to study also systems with delays in the state [13,29] and in more general frameworks, such as for stochastic control systems [19] or fractional integro-differential systems [2]. The main idea of relative controllability is that, instead of controlling the state x t : [−r, 0] → C d of (1.1), defined by x t (s) = x(t + s), in a certain function space such as C k ([−r, 0], C d ) or L p ((−r, 0), C d ), where r ≥ max j∈ 1,N Λ j , one controls only the final state x(t) = x t (0).…”

Section: Introductionmentioning

confidence: 99%

Relative Controllability of Linear Difference Equations

Mazanti¹

2017

SIAM J. Control Optim.

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In this paper, we study the relative controllability of linear difference equations with multiple delays in the state by using a suitable formula for the solutions of such systems in terms of their initial conditions, their control inputs, and some matrix-valued coefficients obtained recursively from the matrices defining the system. Thanks to such formula, we characterize relative controllability in time T in terms of an algebraic property of the matrix-valued coefficients, which reduces to the usual Kalman controllability criterion in the case of a single delay. Relative controllability is studied for solutions in the set of all functions and in the function spaces L p and C k . We also compare the relative controllability of the system for different delays in terms of their rational dependence structure, proving that relative controllability for some delays implies relative controllability for all delays that are "less rationally dependent" than the original ones, in a sense that we make precise. Finally, we provide an upper bound on the minimal controllability time for a system depending only on its dimension and on its largest delay.Notations In this paper, we denote by N and N * the sets of nonnegative and positive integers, respectively. For a, b ∈ R, we write the set of all integers between a and b as a,The cardinality of a set N is denoted by #N. For ξ ∈ R N , we use ξ min and ξ max to denote the smallest and the largest components of ξ , respectively. For ξ ∈ R, the symbol ⌊ξ ⌋ is used to the denote the integer part of ξ , i.e., the unique integer such that ξ − 1 < ⌊ξ ⌋ ≤ ξ .The set of d × m matrices with coefficients in

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Section: Introductionmentioning

confidence: 99%

Relative Controllability of Linear Difference Equations

Mazanti¹

2017

SIAM J. Control Optim.

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“…Relative controllability and its related problems of linear systems represented by different type delay systems have been studied in literature . In particular, rank and Kalman criteria for relative controllability are studied extensively.…”

Section: Introductionmentioning

confidence: 99%

Finite time stability and relative controllability of Riemann‐Liouville fractional delay differential equations

Wang

2019

Math Methods in App Sciences

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This paper firstly deals with finite time stability (FTS) of Riemann-Liouville fractional delay differential equations via giving a series of properties of delayed matrix function of Mittag-Leffler. We secondly study relative controllability of such type-controlled system. With the help of the representation of solution, both Gram-like type matrix and rank criterion are derived, which extend the corresponding results for linear systems. KEYWORDSdelayed matrix function of Mittag-Leffler, finite time stability, fractional delay differential equations, relative controllability INTRODUCTIONRecently, some authors apply delayed exponential matrix function to find the explicit solution formula of linear delay equations. 1-3 Further, there are some continued work on finding the representation of solutions to delay continuous and discrete systems; see, for example, previous studies. [4][5][6][7][8][9][10][11] Fractional differential equations have been used in solving the practical problems in the fields of natural science. 12,13 In particular, FTS of linear fractional delay differential equations with controls was offered in Lazarević, 14 and an inequality of Gronwall type is used to derive the results. For other related contribution, one can refer to previous works. [15][16][17][18][19][20][21][22][23][24][25] However, an initial singular condition for the Riemann-Liouville fractional differential equations is much different from the standard initial condition for a Caputo fractional differential equations. The main results of the standard initial condition for a Caputo fractional delay differential equation were obtained in Li and Wang. 11 Relative controllability and its related problems of linear systems represented by different type delay systems have been studied in literature. [26][27][28][29][30][31][32][33] In particular, rank and Kalman criteria for relative controllability are studied extensively.Very recently, Li and Wang 12 studiedHere, D − + x denotes Riemann-Liouville fractional derivative, I 1− − + x denotes Riemann-Liouville fractional integral, T = k * < ∞ for a fixed k * ∈ {1, 2, … }, is a constant, ∈ C([− , T], R n ), and (D − + )( ) exists. We consider the Riemann-Liouville fractional derivative D − + , where the lower limit is − + (− is a singular point). To keep the "consistence" with the lower limit in Riemann-Liouville derivative, we keep the initial condition at − + .

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“…Due to the infinite-dimensional nature of the dynamics of difference equations and neutral functional differential equations, several different notions of controllability can be used, such as approximate, exact, spectral, or relative controllability [34,5,12,31,24]. Relative controllability was originally introduced in the study of control systems with delays in the control input [5] and consists in controlling the value of x(T ) ∈ C d at some prescribed time T .…”

Section: Introductionmentioning

confidence: 99%

“…Relative controllability was originally introduced in the study of control systems with delays in the control input [5] and consists in controlling the value of x(T ) ∈ C d at some prescribed time T . In the context of difference equations of the form (1.1), it was characterized in some particular situations with integer delays in [12,31], with a complete characterization on the general case provided in [24].…”

Section: Introductionmentioning

confidence: 99%