Final Report Summary - COVMAPS (Covering mappings and their applications in functional equations, difference equations and optimization)
COVMAPS - Covering mappings and their applications in functional equations, difference equations and optimization
The research program of the COVMAPS project concerns covering mappings and their applications. The key research effort addresses the properties of covering mappings in generalized metric spaces as well as sufficient solvability conditions for multiple types of equations and inclusions defined by conditionally covering multi-valued mappings in metric spaces. These technical results were applied to the investigation of solvability conditions, existence of equilibrium, and stability conditions of types of equations and inclusions with practical significance in many application areas. We single out difference equations and several types of functional equations. The research on the later will be based on results obtained for covering mappings in generalized metric spaces. These results will be the basis to address the technical challenges arising in the research of global solvability conditions for control systems, as well as, necessary optimality conditions for control systems defined by Volterra equations.
The results of the COVMAPS project are in the realm of Mathematical Analysis with relevance to control and optimization, and, therefore, they are of interest to the Mathematical and Engineering research communities. Their pertinence is strengthened by the challenges arising in the rapid development of the Information Society Technologies as they require increasingly sophisticated Mathematical frameworks to support the design of advanced systems. Key classes of problems concern the control and optimization of hybrid and impulsive systems to model networked interacting or distributed agents, which encompass both the so called “cyber-physical systems” and also the “systems of systems”. This emergent paradigm has been increasingly regarded as the most promising for large scale, distributed, networked systems.
The scientific objectives of the project are organized as follows:
Task 1 – Sufficient conditions for the composition of two locally covering single-valued or set-valued mappings to be locally covering.
Task 2 – Sufficient conditions for single-valued locally covering mapping to have a continuous inverse mapping.
Task 3 – Conditions for Lipschitz perturbed conditionally covering set-valued mappings to have a covering property.
Task 4 – Properties of covering mappings in generalized metric spaces, including the research of their topological properties, to obtain global and local coincidence points theorems, and covering properties of certain mappings on generalized metric spaces.
Task 5 – Global solvability conditions for controlled differential systems defined by functional-differential equation and subject to mixed constraints.
Task 6 – Solvability conditions for difference equations in metric spaces difference equations, including conditions for the existence of equilibria, and conditions for their stability.
Task 7 – Solvability conditions for equations in functional generalized metric spaces.
Task 8 – First order optimality conditions for an optimal control problem of generalized type in which ordinary differential dynamics will be replaced by a Volterra equation.
The obtained results pertain all the Tasks, with a weaker emphasis in Task 8, were published in high standard peer-reviewed journals, conferences, and reports, as well as other reputed scientific events. Briefly, these consist in:
a) Sufficient conditions ensuring the covering property of the composition of two set-valued mappings in terms of covering mappings on a collection of balls. Moreover, covering criteria for a certain type of smooth mappings were obtained.
b) Relations between the most relevant concepts of covering and metric regularity in the literature were established.
c) A theorem on the Lipschitz perturbation of conditionally covering set-valued mappings providing sufficient conditions for solvability of certain inclusions was proved and applied to the Cauchy problem for differential inclusions to obtain local solvability conditions and solution estimates.
d) Both local and global types of covering for mappings in generalized metric spaces were investigated. Coincidence points theorems for single and set valued mappings acting in generalized metric spaces, and the stability of their coincidence points were proved. In the sequel, estimates for distances from arbitrary point to the intersection of graphs of two set-valued mappings and for the Hausdorff distance of their coincidence points were obtained.
e) Solvability conditions for control systems with control constraints and mixed (i.e. joint state and control) constraints were derived when: (i) the mixed constraints function is smooth, and the control constraint set is a closed convex cone, and (ii) the mixed constraint function is not smooth but it is covering with respect to the control variable, and the control constraint set is just a closed set. In the sequel, solvability conditions for control systems with mixed constraints and implicit dynamics were obtained together with the existence of admissible continuous controls and solution estimates.
f) Properties of some functional generalized metric spaces were investigated to obtain results on coincidence points of mappings acting in generalized metric space which were applied to obtain solvability conditions for functional equations in the space of continuous functions.
g) Existence of solutions to implicit difference equations under very weak assumptions were obtained by applying the derived covering mappings results which also enabled the investigation of their asymptotic behaviors including the existence of stable equilibrium.
h) The concept of covering for mappings and multi-valued mappings in partially ordered spaces is introduced and sufficient conditions for the existence of coincidence points and minimal coincidence points of isotone and orderly covering mappings are obtained. These results generalize classical fixed point theorems for isotone mappings. Moreover, the known theorems on coincidence points of covering and Lipschitz mappings in metric spaces are deduced from the obtained results.
j) Sufficient conditions for the existence of an equilibrium price vector and its stability with respect to small perturbations in a nonlinear market model are given based on the existence and stability of coincidence points in the theory of covering mappings.
l) Sufficient conditions for the continuous, Holder continuous, or Lipschitz continuous dependence of a coincidence point on a parameter are obtained. As a corollary, a very general implicit function theorem for an inclusion defined by a Frechet differentiable mapping in Banach spaces is obtained.
k) Sufficient conditions for the covering property of maps of spaces of compact sets and for the existence of coincidence sets of maps of metric spaces are obtained.
Besides being a solid contribution to the construction of a key body of results in Mathematical Analysis, the results obtained in this project contribute to the widening of the practical applicability of control and optimization theories to the various classes of systems such as those arising in socioeconomics, finance, robotics, manufacturing, biological and environmental domains. The results to be derived will contribute to a more sophisticated knowledge of extremum conditions for such problems usually leads not only to the synthesis of better strategies, but also to a better understanding of the control or optimization problem thus enabling the formulation of a more adequate approach (problem formulation, numerical tools, etc.) which, in turn, enables a more precise and relevant solution.
The Institute for System and Robotics Porto hosting the COVMAPS project has a very close cooperative relationship with the LSTS - Laboratory for Underwater Systems and Technologies – of FEUP which has a worldwide recognized field experience with networked unmanned robotic vehicles whose on-board computers run optimization based control algorithms. Although a number of intermediate research stages are still required, LSTS may prove to be a future testing ground for algorithms enhanced by some of the COVMAPS pioneer results.
COVMAPS website: http://paginas.fe.up.pt/~sergey/covmaps1.htm
The research program of the COVMAPS project concerns covering mappings and their applications. The key research effort addresses the properties of covering mappings in generalized metric spaces as well as sufficient solvability conditions for multiple types of equations and inclusions defined by conditionally covering multi-valued mappings in metric spaces. These technical results were applied to the investigation of solvability conditions, existence of equilibrium, and stability conditions of types of equations and inclusions with practical significance in many application areas. We single out difference equations and several types of functional equations. The research on the later will be based on results obtained for covering mappings in generalized metric spaces. These results will be the basis to address the technical challenges arising in the research of global solvability conditions for control systems, as well as, necessary optimality conditions for control systems defined by Volterra equations.
The results of the COVMAPS project are in the realm of Mathematical Analysis with relevance to control and optimization, and, therefore, they are of interest to the Mathematical and Engineering research communities. Their pertinence is strengthened by the challenges arising in the rapid development of the Information Society Technologies as they require increasingly sophisticated Mathematical frameworks to support the design of advanced systems. Key classes of problems concern the control and optimization of hybrid and impulsive systems to model networked interacting or distributed agents, which encompass both the so called “cyber-physical systems” and also the “systems of systems”. This emergent paradigm has been increasingly regarded as the most promising for large scale, distributed, networked systems.
The scientific objectives of the project are organized as follows:
Task 1 – Sufficient conditions for the composition of two locally covering single-valued or set-valued mappings to be locally covering.
Task 2 – Sufficient conditions for single-valued locally covering mapping to have a continuous inverse mapping.
Task 3 – Conditions for Lipschitz perturbed conditionally covering set-valued mappings to have a covering property.
Task 4 – Properties of covering mappings in generalized metric spaces, including the research of their topological properties, to obtain global and local coincidence points theorems, and covering properties of certain mappings on generalized metric spaces.
Task 5 – Global solvability conditions for controlled differential systems defined by functional-differential equation and subject to mixed constraints.
Task 6 – Solvability conditions for difference equations in metric spaces difference equations, including conditions for the existence of equilibria, and conditions for their stability.
Task 7 – Solvability conditions for equations in functional generalized metric spaces.
Task 8 – First order optimality conditions for an optimal control problem of generalized type in which ordinary differential dynamics will be replaced by a Volterra equation.
The obtained results pertain all the Tasks, with a weaker emphasis in Task 8, were published in high standard peer-reviewed journals, conferences, and reports, as well as other reputed scientific events. Briefly, these consist in:
a) Sufficient conditions ensuring the covering property of the composition of two set-valued mappings in terms of covering mappings on a collection of balls. Moreover, covering criteria for a certain type of smooth mappings were obtained.
b) Relations between the most relevant concepts of covering and metric regularity in the literature were established.
c) A theorem on the Lipschitz perturbation of conditionally covering set-valued mappings providing sufficient conditions for solvability of certain inclusions was proved and applied to the Cauchy problem for differential inclusions to obtain local solvability conditions and solution estimates.
d) Both local and global types of covering for mappings in generalized metric spaces were investigated. Coincidence points theorems for single and set valued mappings acting in generalized metric spaces, and the stability of their coincidence points were proved. In the sequel, estimates for distances from arbitrary point to the intersection of graphs of two set-valued mappings and for the Hausdorff distance of their coincidence points were obtained.
e) Solvability conditions for control systems with control constraints and mixed (i.e. joint state and control) constraints were derived when: (i) the mixed constraints function is smooth, and the control constraint set is a closed convex cone, and (ii) the mixed constraint function is not smooth but it is covering with respect to the control variable, and the control constraint set is just a closed set. In the sequel, solvability conditions for control systems with mixed constraints and implicit dynamics were obtained together with the existence of admissible continuous controls and solution estimates.
f) Properties of some functional generalized metric spaces were investigated to obtain results on coincidence points of mappings acting in generalized metric space which were applied to obtain solvability conditions for functional equations in the space of continuous functions.
g) Existence of solutions to implicit difference equations under very weak assumptions were obtained by applying the derived covering mappings results which also enabled the investigation of their asymptotic behaviors including the existence of stable equilibrium.
h) The concept of covering for mappings and multi-valued mappings in partially ordered spaces is introduced and sufficient conditions for the existence of coincidence points and minimal coincidence points of isotone and orderly covering mappings are obtained. These results generalize classical fixed point theorems for isotone mappings. Moreover, the known theorems on coincidence points of covering and Lipschitz mappings in metric spaces are deduced from the obtained results.
j) Sufficient conditions for the existence of an equilibrium price vector and its stability with respect to small perturbations in a nonlinear market model are given based on the existence and stability of coincidence points in the theory of covering mappings.
l) Sufficient conditions for the continuous, Holder continuous, or Lipschitz continuous dependence of a coincidence point on a parameter are obtained. As a corollary, a very general implicit function theorem for an inclusion defined by a Frechet differentiable mapping in Banach spaces is obtained.
k) Sufficient conditions for the covering property of maps of spaces of compact sets and for the existence of coincidence sets of maps of metric spaces are obtained.
Besides being a solid contribution to the construction of a key body of results in Mathematical Analysis, the results obtained in this project contribute to the widening of the practical applicability of control and optimization theories to the various classes of systems such as those arising in socioeconomics, finance, robotics, manufacturing, biological and environmental domains. The results to be derived will contribute to a more sophisticated knowledge of extremum conditions for such problems usually leads not only to the synthesis of better strategies, but also to a better understanding of the control or optimization problem thus enabling the formulation of a more adequate approach (problem formulation, numerical tools, etc.) which, in turn, enables a more precise and relevant solution.
The Institute for System and Robotics Porto hosting the COVMAPS project has a very close cooperative relationship with the LSTS - Laboratory for Underwater Systems and Technologies – of FEUP which has a worldwide recognized field experience with networked unmanned robotic vehicles whose on-board computers run optimization based control algorithms. Although a number of intermediate research stages are still required, LSTS may prove to be a future testing ground for algorithms enhanced by some of the COVMAPS pioneer results.
COVMAPS website: http://paginas.fe.up.pt/~sergey/covmaps1.htm