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Öğe On the robustness of the integrable trajectories of the control systems with limited control resources(Polish Academy of Sciences, 14.09.2023) Hüseyin, Nesir; Hüseyin, Anar; Guseinov, KhalikThe control system described by Urysohn type integral equation is considered where the system is nonlinear with respect to the phase vector and is affine with respect to the control vector. The control functions are chosen from the closed ball of the space $L_q\left(\Omega;\mathbb{R}^m\right),$ $q>1,$ with radius $r$ and centered at the origin. The trajectory of the system is defined as $p$-integrable multivariable function from the space $L_p\left(\Omega;\mathbb{R}^n\right),$ $\frac{1}{q}+\frac{1}{p}=1,$ satisfying the system's equation almost everywhere. It is shown that the system's trajectories are robust with respect to the fast consumption of the remaining control resource. Applying this result it is proved that every trajectory can be approximated by the trajectory obtained by full consumption of the total control resource.Öğe On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources(Celal Bayar University, 30.06.2023) Hüseyin, AnarIn this paper the control system with integral constraint on the control functions is studied where the behavior of the system by the Urysohn type integral equation is described. The admissible control functions are chosen from the closed ball of the space ... .Öğe On the robustness of continuous trajectories of the control system described by an integral equation(KOREAN SOC MATHEMATICAL EDUCATION, 30.05.2023) Hüseyin, Nesir; Hüseyin, AnarIn this paper the control system described by Urysohn type integral equation is studied. It is assumed that control functions are integrally constrained. The trajectory of the system is defined as multivariable continuous function which satisfies the system’s equation everywhere. It is shown that the set of trajectories is Lipschitz continuous with respect to the parameter which characterizes the bound of the control resource. An upper estimation for the diameter of the set of trajectories is obtained. The robustness of the trajectories with respect to the fast consumption of the remaining control resource is discussed. It is proved that every trajectory can be approximated by the trajectory obtained by full consumption of the control resource.Öğe On the Vietoris semicontinuity property of the $L_p$ balls at $p=1$ and an application(Birkhauser Verlag, 19.07.2023) Hüseyin, AnarIn this paper the Vietoris right lower semicontinuity at $p=1$ of the set valued map $p\rightarrow B_{\Omega,\mathcal{X},p}(r)$, $p\in [1,\infty]$, is discussed where $B_{\Omega,\mathcal{X},p}(r)$ is the closed ball of the space $L_{p}(\Omega,\Sigma,\mu; \mathcal{X})$ centered at the origin with radius $r$, $(\Omega,\Sigma,\mu)$ is a finite and positive measure space, $\mathcal{X}$ is a separable Banach space. It is proved that the considered set valued map is Vietoris right lower semicontinuous at $p=1$. Introducing additional geometric constraints on the functions from the ball $B_{\Omega,\mathcal{X},1}(r)$, a property which is close to the Hausdorff right lower semicontinuity, is derived. An application of the obtained result to the set of integrable outputs of the input-output system described by the Urysohn type integral operator is studied.Öğe Approximations of the set of trajectories and integral funnel of the nonlinear control systems with $L_p$ norm constraints on the control functions(Oxford University Press, 29.11.2022) Hüseyin, AnarIn this paper, approximations of the set of trajectories and integral funnel of the control system described by non-linear ordinary differential equation with integral constraint on the control functions are considered. The set of admissible control functions is replaced by a set, consisting of a finite number of piecewise-constant control functions. It is shown that the set of trajectories generated by a finite number of piecewise-constant control functions is an internal approximation of the set of trajectories. Further, each trajectory generated by a piecewise-constant control function is substituted by appropriate Euler’s broken line and it is proved that the set consisting of a finite number of Euler’s broken lines is an approximation of the set of trajectories of given control system. An approximation of the system’s integral funnel by a set consisting of a finite number of points is obtained.Öğe On the p-integrable trajectories of the nonlinear control system described by Urysohn type integral equation(Walter de Gruyter GmbH, 11.10.2022) Hüseyin, AnarThe control systems described by the Urysohn-type integral equations and integral constraints on the control functions are considered. The functions from the closed ball of the space $L_p$, $p>1$, with radius $r$, are chosen as admissible control functions. The trajectory of the system is defined as a $p$-integrable function, satisfying the system’s equation almost everywhere. The boundedness and path-connectedness of the set of $p$-integrable trajectories are discussed. It is illustrated that the set of trajectories, in general, is not a closed subset of the space $L_p$. The robustness of a trajectory with respect to the fast consumption of the remaining control resource is established, and it is proved that every trajectory of the system can be approximated by the trajectory obtained by the full consumption of the control resource.
