scholarly journals On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3288
Author(s):  
Alexey Iskakov ◽  
Igor Yadykin
Keyword(s):  
Existence Of Solutions ◽  
Bilinear System ◽  
Lyapunov Equation ◽  
Existence Of A Solution ◽  
Bilinear Control ◽  
Spectral Decompositions ◽  
Generalized Lyapunov Equation ◽  
Controllability And Observability ◽  
Bilinear Equations ◽  
Solution Matrix

The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be divided into parts (sub-Gramians) that characterize the measure of these generalized modes and their interactions. Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability and observability. We also propose an algorithm for computing the Gramians and sub-Gramians based on the element-wise computation of the solution matrix. Based on the proposed algorithm, a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed, which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear equations. Examples are provided that illustrate the application and practical use of the considered spectral decompositions.

scholarly journals On partial fractional Sturm–Liouville equation and inclusion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zohreh Zeinalabedini Charandabi ◽  
Hakimeh Mohammadi ◽  
Shahram Rezapour ◽  
Hashem Parvaneh Masiha
Keyword(s):  
Differential Equation ◽  
Liouville Equation ◽  
Existence Of Solutions ◽  
Liouville Problem ◽  
Existence Of A Solution ◽  
Contractive Mappings ◽  
Sturm Liouville

AbstractThe Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

On the discrete generalized Lyapunov equation

Automatica ◽  
1995 ◽  
Vol 31 (2) ◽  
pp. 297-301 ◽  
Author(s):  
Vassilis L. Syrmos ◽  
Pradeep Misra ◽  
Ravi Aripirala
Keyword(s):  
Lyapunov Equation ◽  
Generalized Lyapunov Equation

On a semilinear equation in ℝ2 involving bounded measures

1983 ◽  
Vol 95 (3-4) ◽  
pp. 181-202 ◽  
Author(s):  
Juan L. Vazquez
Keyword(s):  
Real Function ◽  
Radon Measure ◽  
Existence Of Solutions ◽  
Critical Value ◽  
Existence Of A Solution ◽  
One Dimensional ◽  
Semilinear Equation ◽  
The One

SynopsisWe study the semilinear equation –Δu + β(u) = f in ℝ2, where β is a continuous increasing real function with β(0) = 0 and f is a bounded Radon measure. We show the existence of a solution, which is unique in the appropriate class, provided that each of the point masses contained in f does not exceed some critical value denned in terms of the growth of (β at ∞ This condition is shown to be necessary for the existence of solutions, even locally. The one-dimensional situation is also discussed.

scholarly journals SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HÖLDER-α DOMAINS

2010 ◽  
Vol 20 (01) ◽  
pp. 95-120 ◽  
Author(s):  
RICARDO G. DURÁN ◽  
FERNANDO LÓPEZ GARCÍA
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Basic Result ◽  
Weighted Sobolev Spaces ◽  
Mean Value ◽  
Well Posedness ◽  
Existence Of A Solution ◽  
Simply Connected ◽  
Distance To The Boundary

If Ω ⊂ ℝn is a bounded domain, the existence of solutions [Formula: see text] of div u = f for f ∈ L2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution [Formula: see text], where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution [Formula: see text] for some r < 2 depending on the power of the cusp.

Study of Controllability of Bilinear Systems with Limited Control

2021 ◽  
Vol 26 (3-4) ◽  
pp. 302-313
Author(s):  
L.G. Gagarina ◽  
◽  
A.A. Doronina ◽  
R.A. Fomin ◽  
D.A. Chukhlyaev ◽  
...  
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Control Systems ◽  
Control Program ◽  
Bilinear System ◽  
Equation System ◽  
Bilinear Systems ◽  
Complete Controllability ◽  
Complex Objects ◽  
Bilinear Control

Optimal control is closely related to the choice of the most advantageous control modes for complex objects, which are described using ordinary differential systems. The problem of optimal control consists in calculating the optimal control program and synthesizing the optimal control system. This problem arises in the applied field of the optimal control theory, in the case when control is based on the principle of feedback and in automatic control systems. Optimal control problems, as a rule, are calculated by numerical methods to find the extremum of a functional or to solve a boundary value problem for a differential equation system. From a mathematical standpoint, the synthesis of optimal control systems is a nonlinear programming problem in functional spaces. In this study the problem of complete controllability of a bilinear control system on the plane was considered. The controllability of bilinear systems with both unlimited and limited control was studied. The evidences of closed trajectory systems controllability theorems were produced. The authors have defined multiple criteria of complete controllability for bilinear system with limited control. The complete controllability conditions of bilinear control system have been proposed with their algebraic reasoning. In the contemporary context of universal robotization of production, completely controllable systems matter in navigation, as well as in modeling of a number of economic and social processes.

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