English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we study the optimal control problem with terminal andinequality state constraints for state equations described by Volterra integralequations having singular and nonsingular kernels. The singular kernelintroduces abnormal behavior of the state trajectory with respect to theparameter of $\alpha \in (0,1)$. Our state equation is able to cover variousstate dynamics such as any types of Volterra integral equations withnonsingular kernels only, fractional differential equations (in the sense ofRiemann-Liouville or Caputo), and ordinary differential state equations. Weobtain the well-posedness (in $L^p$ and $C$ spaces) and precise estimates ofthe state equation using the generalized Gronwall's inequality and the properregularities of integrals having singular and nonsingular integrands. We thenprove the maximum principle for the corresponding state-constrained optimalcontrol problem. In the derivation of the maximum principle, due the presenceof the state constraints and the control space being only a separable metricspace, we have to employ the Ekeland variational principle and the spikevariation technique, together with the intrinsic properties of distancefunctions and the generalized Gronwall's inequality, to obtain the desirednecessary conditions for optimality. In fact, as the state equation has bothsingular and nonsingular kernels, the maximum principle of this paper is new,where its proof is more involved than that for the problems of Volterraintegral equations studied in the existing literature. Examples are provided toillustrate the theoretical results of this paper.

Maximum Principle for State-Constrained Optimal Control Problems of Volterra Integral Equations having Singular and Nonsingular Kernels