{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/116067"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/116067","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Numerical Estimation of L2 Gain for Nonlinear Input-Output Systems","abstract":"The L2 gain of a nonlinear time-dependent system measures the maximal gain in the transfer of energy from admissible input signals to the output signals, in which both the input and output signals are measured with the L2 norm. For general nonlinear systems, obtaining a sharp estimate of the L2 gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of L2 gains for systems with quadratic nonlinearity. The approach utilizes a recently developed method that solves a class of Hamilton-Jacobi-Bellman equations via a Taylor series-based approximation, which is scalable to high-dimensional problems given the utilization of linear tensor systems. The ideas are demonstrated through a few concrete examples that include a one-dimensional problem with an explicit energy function and several Galerkin approximations of the viscous Burgers equation.","abstract_html":"The L2 gain of a nonlinear time-dependent system measures the maximal gain in the transfer of energy from admissible input signals to the output signals, in which both the input and output signals are measured with the L2 norm. For general nonlinear systems, obtaining a sharp estimate of the L2 gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of L2 gains for systems with quadratic nonlinearity. The approach utilizes a recently developed method that solves a class of Hamilton-Jacobi-Bellman equations via a Taylor series-based approximation, which is scalable to high-dimensional problems given the utilization of linear tensor systems. The ideas are demonstrated through a few concrete examples that include a one-dimensional problem with an explicit energy function and several Galerkin approximations of the viscous Burgers equation.","abstract_has_math":false,"creators":["Lang, Sydney"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Borggaard, Jeffrey T.","Liu, Honghu"],"committee_members":["Ciupe, Stanca M."],"year":2023,"date_issued":"2023-08-21","date_published":"2023-08-21","updated_at":"2026-07-22T22:20:31Z","subjects":["L2 gain","nonlinear systems","input-output systems"],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:38397"],"render_values":[{"text":"vt_gsexam:38397","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/116067","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Borggaard, Jeffrey T.","Liu, Honghu"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Ciupe, Stanca M."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Lang, Sydney"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2023-08-22T08:00:12Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2023-08-22T08:00:12Z"]},{"key":"dc:date.issued","label":"Date","values":["2023-08-21"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["L2 gain","nonlinear systems","input-output systems"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:38397"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/116067"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The L2 gain of a nonlinear time-dependent system measures the maximal gain in the transfer of energy from admissible input signals to the output signals, in which both the input and output signals are measured with the L2 norm. For general nonlinear systems, obtaining a sharp estimate of the L2 gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of L2 gains for systems with quadratic nonlinearity. The approach utilizes a recently developed method that solves a class of Hamilton-Jacobi-Bellman equations via a Taylor series-based approximation, which is scalable to high-dimensional problems given the utilization of linear tensor systems. The ideas are demonstrated through a few concrete examples that include a one-dimensional problem with an explicit energy function and several Galerkin approximations of the viscous Burgers equation."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["With nonlinear systems that are of the form of input-output models, questions often arise as to how to measure the energy that passes through such systems and determine strategies to look for specific signals that allow the designer freedom to explore certain system behaviors. The energy comes in the form of a signal. For general nonlinear systems, obtaining a sharp estimate of such energy gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of these gains for systems with quadratic nonlinearity. The approach combines fundamental theoretical understandings established in the literature with scalable software recently developed in approximating the solution of the underlying partial differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation. In this approach, the energy gain is linked to a single scalar parameter in the HJB equation. Roughly speaking, the energy gain is the lower bound of this scalar parameter above which the HJB equation always admits a non-negative solution. Thus, it boils down to approximating the HJB solution using the software while changing this scalar parameter. We will present the theoretical foundation of the approach and illustrate the foundation through several academic examples ranging from low to relatively high dimensions."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Numerical Estimation of L2 Gain for Nonlinear Input-Output Systems"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Borggaard, Jeffrey T.","Liu, Honghu"],"dc:contributor.committeemember":["Ciupe, Stanca M."],"dc:contributor.department":["Mathematics"],"dc:creator":["Lang, Sydney"],"dc:date.accessioned":["2023-08-22T08:00:12Z"],"dc:date.available":["2023-08-22T08:00:12Z"],"dc:date.issued":["2023-08-21"],"dc:description.abstract":["The L2 gain of a nonlinear time-dependent system measures the maximal gain in the transfer of energy from admissible input signals to the output signals, in which both the input and output signals are measured with the L2 norm. For general nonlinear systems, obtaining a sharp estimate of the L2 gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of L2 gains for systems with quadratic nonlinearity. The approach utilizes a recently developed method that solves a class of Hamilton-Jacobi-Bellman equations via a Taylor series-based approximation, which is scalable to high-dimensional problems given the utilization of linear tensor systems. The ideas are demonstrated through a few concrete examples that include a one-dimensional problem with an explicit energy function and several Galerkin approximations of the viscous Burgers equation."],"dc:description.abstractgeneral":["With nonlinear systems that are of the form of input-output models, questions often arise as to how to measure the energy that passes through such systems and determine strategies to look for specific signals that allow the designer freedom to explore certain system behaviors. The energy comes in the form of a signal. For general nonlinear systems, obtaining a sharp estimate of such energy gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of these gains for systems with quadratic nonlinearity. The approach combines fundamental theoretical understandings established in the literature with scalable software recently developed in approximating the solution of the underlying partial differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation. In this approach, the energy gain is linked to a single scalar parameter in the HJB equation. Roughly speaking, the energy gain is the lower bound of this scalar parameter above which the HJB equation always admits a non-negative solution. Thus, it boils down to approximating the HJB solution using the software while changing this scalar parameter. We will present the theoretical foundation of the approach and illustrate the foundation through several academic examples ranging from low to relatively high dimensions."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:38397"],"dc:identifier.uri":["http://hdl.handle.net/10919/116067"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["L2 gain","nonlinear systems","input-output systems"],"dc:title":["Numerical Estimation of L2 Gain for Nonlinear Input-Output Systems"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:31Z"}