We present a method for improving frequency stability in a self-sustained oscillator using a secondary feedback loop acting outside the main sustaining loop with adaptively controlled amplitude and phase shift. We show that quite simple adaptive control laws for the control states of the secondary feedback signal can affect the way that noise circulates in the oscillator so as to reduce or even eliminate (in theory) phase diffusion, i.e., the rate of linear growth with time of the variance of the oscillator output phase. Our rigorous treatment of this effect is based on a linearized analysis of the noisy slow-flow amplitude and phase equations for a relevant general class of systems, from which we derive an explicit expression for the corresponding asymptotic rate of phase diffusion. Using this result, we consider different choices for tailoring the actuation from the secondary feedback loop. This includes tuning a phase coupling constant that ensures that the noise driving the output phase also drives one of the adaptive control states, which represents a generalization of a desirable behavior observed in internally resonant coupled mode operation. We show that such feedback design may be used to eliminate effects that commonly arise from the conversion of amplitude fluctuations to phase diffusion, analogous to operation at zero dispersion points, and to improve upon the phase-cleaning effect associated with internal resonance to even achieve the ideal situation of zero phase diffusion. We validate our theoretical findings with numerical simulations that agree remarkably well with the theory. The presented results establish a framework for achieving extraordinary frequency stability using slowly varying states of the resonator and feedback.

This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near normally hyperbolic attracting periodic orbits (limit cycles) and quasiperiodic invariant tori of the deterministic dynamical systems obtained in the absence of noise. The formulation uses adjoints to construct a continuous family of transversal hyperplanes that are invariant under the linearized deterministic flow near the limit cycle or quasiperiodic invariant torus. The intersections with each hyperplane of stochastic trajectories that remain near the deterministic cycle or torus over intermediate times may be approximated by a Gaussian distribution whose covariance matrix can be obtained from the solution to the corresponding boundary-value problem. In the case of limit cycles, the analysis improves upon results in the literature through the explicit use of state-space projections, transversality constraints, and symmetry-breaking parameters that ensure uniqueness of the solution despite the lack of hyperbolicity along the limit cycle. These same innovations are then generalized to the case of a quasiperiodic invariant torus of arbitrary dimension. In each case, a closed-form solution to the covariance boundary-value problem is found in terms of a convergent series. The methodology is validated against the results of numerical integration for two examples of stochastically perturbed limit cycles and one example of a stochastically perturbed two-dimensional quasiperiodic invariant torus in R2, R2 \times S1, and R2 \times S1, respectively, for which explicit expressions may be found for the associated covariance functions using the proposed series solutions. Finally, an implementation of the covariance boundary-value problem in the numerical continuation package coco is applied to analyze the small-noise limit near a two-dimensional quasiperiodic invariant torus in a nonlinear deterministic dynamical system in R4 that does not support closed-form analysis. Excellent agreement with numerical evidence from stochastic time integration shows the potential for using deterministic continuation techniques to study the influence of stochastic perturbations for both autonomous and periodically excited deterministic vector fields.

Bifurcation analysis collects techniques for characterizing the dependence of certain classes of solutions of a dynamical system on variations in problem parameters. Common solution classes of interest include equilibria and periodic orbits, the number and stability of which may vary as parameters vary. Continuation techniques generate continuous families of such solutions in the combined state and parameter space, e.g., curves (branches) of periodic orbits or surfaces of equilibria. Their advantage over simulation-based approaches is the ability to map out such families independently of the dynamic stability of the equilibria or periodic orbits. Bifurcation diagrams represent families of equilibria and periodic orbits as curves or surfaces in appropriate coordinate systems. Special points, such as bifurcations, are often highlighted in such diagrams. This article provides an illustration of this paradigm of synergy between theoretical derivations and computational analysis for several characteristic examples of bifurcation analysis in commonly encountered classes of problems. General theoretical principles are deduced from these illustrations and collected for the reader's subsequent reference.

In this work, we analyze a mass sensing mechanism that relies on self-excited template dynamics of a simple network of two coupled oscillators, one linear and one nonlinear, to detect changes in the mass of the linear oscillator with tunable sensitivity. The resultant shift in the ratio of oscillation amplitudes is predicted using perturbation analysis and verified against numerical results obtained via parameter continuation. A physical realization of the mass sensing mechanism is proposed, in which the linear oscillator is represented by a periodically excited actuator/microcantilever assembly, the nonlinear oscillator is simulated in silico on a finite interval of time while driven by the periodic steady-state response of the linear oscillator, and a nonlinear root-finding algorithm is used to identify the input to the actuator that reproduces the sought, coupled, self-excited dynamics. The analysis shows that the sensor gain can be adjusted by appropriate tuning of the coupling stiffness without modifications to the physical components. Numerical results for two different actuator models demonstrate rapid convergence of the root-finding algorithm from a trivial initial solution guess over ranges of values of model parameters and added mass ratios.

This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near transversally stable periodic orbits (limit cycles) and quasiperiodic invariant tori of the deterministic dynamical systems obtained in the absence of noise. The formulation uses adjoints to construct a continuous family of transversal hyperplanes that are invariant under the linearized deterministic flow near the limit cycle or quasiperiodic invariant torus. The intersections with each hyperplane of stochastic trajectories that remain near the deterministic cycle or torus over intermediate times may be approximated by a Gaussian distribution whose covariance matrix can be obtained from the solution to the corresponding boundary-value problem. In the case of limit cycles, the analysis improves upon results in the literature through the explicit use of state-space projections, transversality constraints, and symmetry-breaking parameters that ensure uniqueness of the solution despite the lack of hyperbolicity along the limit cycle. These same innovations are then generalized to the case of a quasiperiodic invariant torus of arbitrary dimension. In each case, a closed-form solution to the covariance boundary-value problem is found in terms of a convergent series. The methodology is validated against the results of numerical integration for two examples of stochastically perturbed limit cycles and one example of a stochastically perturbed two-dimensional quasiperiodic invariant torus. Finally, an implementation of the covariance boundary-value problem in the numerical continuation package coco is applied to analyze the small-noise limit near a two-dimensional quasiperiodic invariant torus in a nonlinear deterministic dynamical system in ℝ^4 that does not support closed-form analysis.

Applied Mechanics Reviews (AMR) was founded in 1948 under the editorship of Lloyd Hamilton Donnell (1895–1997), in whose honor the biennial Lloyd Hamilton Donnell Applied Mechanics Reviews Paper Award was inaugurated in 2014. The founding of the journal, shortly after World War II, filled a void in the review of state-of-the-art research in applied mechanics, occupied before the war by the German periodical Zeitschrift für Mechanik. The inaugural volume included both Theodore von Karman and Stephen Timoshenko as Editorial Advisors.Over the years, the scope and purpose of AMR evolved. This reflected the changing nature of scientific research in the engineering sciences, the diversification of journal publications, and the development of alternative mechanisms for disseminating research among geographically distributed groups. Today, the journal aims to provide long-shelf-life, state-of-the-art survey articles, and retrospective reviews across all relevant subdisciplines of applied mechanics and engineering science, including fluid and solid mechanics, heat transfer, dynamics and vibration, and applications. Importantly, papers published in AMR emphasize added value beyond what is available in the existing literature. They do so through authoritative commentary and original synthesis, relating and contrasting the authors' original contributions to those of the community.Since 2014, a handful of AMR issues have been dedicated to collaboration with other ASME technical journals to feature significant contributions for a specific discipline in a single issue. Such special issues have aimed to bring to the fore topics of interest to the broader community as well as experts in the discipline, thereby inviting cross-disciplinary collaboration and welcoming new entrants to the field. Past collaborations include with the ASME Journal of Pressure Vessel Technology and the ASME Journal of Vibration and Acoustics in 2014; the ASME Journal of Tribology in 2017; the ASME Journal of Mechanisms and Robotics in 2018; and the ASME Journal of Computational and Nonlinear Dynamics in 2019.The November 2022 and January 2023 issues of AMR constitute the final installment in the series of such collaborations overseen by Harry Dankowicz, who finished his term as Editor-in-Chief in September 2022. These issues developed together with the ASME Journal of Electrochemical Energy Conversion and Storage (JEECS) feature four review papers exemplifying state-of-the-art research in the mechanics of electrochemical energy conversion and storage. A companion Special Section on the Mechanics of Electrochemical Energy Conversion and Storage appeared in the November 2021 issue of JEECS and collected eight shorter papers reporting on original technical research with a similar focus [1].Originally founded in 2004 as the ASME Journal of Fuel Cell Science and Technology and known by its new name since 2016, JEECS serves as a medium for rapid dissemination of original research results focused on processes, components, devices, and systems that store and convert electrical and chemical energy. The journal is a forum for research concerned with advanced materials development, synthesis, manufacturing, and characterization, as well as component, device, and systems design and analysis, with an interest in studies of reliability, durability, and damage tolerance. Areas of application include batteries, fuel cells, electrolyzers, separation membranes, electrochemical capacitors, thermogalvanic cells, and photo-electrochemical cells.The four featured reviews in the November 2022 and January 2023 issues of AMR are all concerned with rechargeable battery technologies. They focus on the interplay of mechanics and electrochemistry in driving a deterioration of battery capacity during repeated cycles of charging and discharging, as well as on phenomena that may induce mechanical and (catastrophic) electrochemical failure.In the work by Gandharapu and Mukhopadhyay (AMR 74(6), November 2022), the authors review causes of stress development and mechanical failure of Sn-based electrodes for future Li-, Na-, and K-ion batteries that rely on cycles of alloying/de-alloying for operation, in contrast to the reversible hosting of Li ions in graphitic carbon-based anodes through intercalation. This review is motivated by growing interest in the use of anode materials with higher storage capacity and reaction potentials well above the Li plating/stripping potential, as well as the use of alternative alkali metals that are more plentiful than lithium, viz., sodium and potassium. The discussion highlights the occurrence during alloying/de-alloying with Sn of sequences of intermetallic phases with different material properties than the parent material. The associated phase transformations produce dramatic volume changes and discontinuities between different material phases that result in stress development, plastic deformation, fracture, and disintegration/pulverization. The paper reviews empirical results on strategies for suppressing the deleterious effects of these phase transformations, including by reducing characteristic length scales to nanoscale dimensions, inserting interlayers of inactive components to accommodate volume changes and maintain conductivity even if the active materials disintegrate, and/or pre-alloying with other inactive or active elements to buffer against dimensional changes and modify reaction pathways. As articulated by the authors, while significant material science developments surely lie in store for these next-generation battery technologies, significant benefits accrue from a fundamental understanding of the coupling between electrochemistry and mechanics.This challenge is explored in greater depth by Gao et al. (AMR 74(6), November 2022), which focuses on anodes made from C, Si, SiOx, and Si/C composite, and reviews the state of the art in theoretical and computational modeling of stress development during electrochemical cycling, as well as experimental characterization of related failure modes. Modes considered in the discussion include cracks on the surface of Si particles, cracks of the carbon shells of Si/C particles that lead to formation of inactive particles after repeated cycling, and cracks within the active layer that destroy the overall electrochemical network; debonding failures between the core and shell of Si/C particles or between active particles and binders that increase interface impedances, and between active layers and the current collector that result in loss of electrical contact; as well as the formation of Li dendrites that may penetrate the solid electrolyte and lead to dangerous short circuits and thermal run-away. The authors further provide a comprehensive survey of stress-management strategies in Si-based high-capacity anodes like those presented for Sn-based anodes by Gandharapu and Mukhopadhyay. In their conclusions, and in describing recent modeling work by the authors and others, they stress the need for multiscale-multiphysics models of the full electrochemomechanical coupling at both particle and electrode levels.The detailed review by Deshpande and McMeeking (AMR 75(1), January 2023) puts an even greater emphasis on efforts to model the combined effects of mechanics, electrochemistry, thermodynamics, and kinetics on phenomena that affect the performance of solid-state batteries with metal anodes. Their discussion provides authoritative commentary on the foundational literature and explores simplified theoretical models and back-of-the-envelope predictions for several relevant phenomena, including the onset of defect-initiated delamination at the interface between the solid electrolyte and a cathode storage particle, crack propagation in the solid electrolyte, and lithium-ion transport in solid electrolytes. The authors review theoretical foundations underlying different reported extensions to the Butler-Volmer equation for redox reactions at the electrolyte/electrode interfaces with emphasis on the influence of mechanical stress. They go on to apply the results to quantitative predictions about possible unstable growth of roughness of a lithium metal anode during charging, as well as the existence of a critical roughness wavelength beyond which growth occurs. A final section investigates models of nucleation and growth of lithium filaments through solid electrolytes and highlights the need to account for the presence of voids at the electrode/electrolyte interface that drive an increased rate of filament growth. In their conclusions, the authors point to several factors requiring further elucidation, including the effects of creep deformation and the processes leading to the formation of large voids.The need for foundational contributions at the intersection of mechanics and electrochemistry is also stressed in the state-of-the-art review by Naik et al. (AMR 75(1), January 2023) on the stability of solid–liquid and solid-solid interfaces in lithium metal batteries. A recurrent theme throughout the discussion is that of heterogeneity—whether in surface morphology, microstructure, material properties, electrochemistry, or percolation pathways, or a result of asymmetric contact loss—and the implications to different failure modes. For example, in a section dedicated to solid state lithium-sulfur batteries, the authors stress the need to consider the effects on transport and reaction kinetics of stress heterogeneity arising from nonuniform lithium-sulfide deposition. Transport and reaction heterogeneities are also called out as playing a critical role in the loss of stability of the solid-electrolyte interphase layer between lithium metal electrodes and liquid electrolyte over multiple charging and discharging cycles. In solid state batteries, surface roughness and microstructural defects are found to cause localized nucleation during lithium plating and contact loss during stripping. Solid-solid point contacts between cathode active materials and solid electrolyte are shown to lead to stress concentrations, current focusing, and localized overpotential. Despite extensive ongoing research into minimizing such heterogeneities, it is clear from this paper that much fundamental work remains.By collecting these papers in two issues of AMR, we hope to bring attention to a fertile area of collaboration between theoretical and experimental mechanicians, material scientists, and electrochemists, one that is vital for modern life and for a future of sustainable technologies and energy utilization. Although the subject matter is presented here at a level of sophistication appropriate for state-of-the-art survey papers, we also hope that some of the content may be used to interest new generations of engineers and scientists in interdisciplinary work grounded in a combination of first principles, cutting-edge experiments, and advanced technologies.

This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the coco software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example coco code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation. 200 words.

This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the coco software package and its explicit support for community-driven development. To this end, the paper first formalizes the coco construction paradigm for augmented continuation problems compatible with simultaneous analysis of implicitly defined manifolds of solutions to nonlinear equations and the corresponding adjoint variables associated with optimization of scalar objective functions along such manifolds. The paper uses applications to data assimilation from finite time histories and phase response analysis of periodic orbits to identify a universal paradigm of construction that permits abstraction and generalization. It then details the theoretical framework for a coco-compatible toolbox able to support the analysis of a large family of delay-coupled multi-segment boundary-value problems, including periodic orbits, quasiperiodic orbits, connecting orbits, initial-value problems, and optimal control problems, as illustrated in a suite of numerical examples. The paper aims to present a pedagogical treatment that is accessible to the novice and inspiring to the expert by appealing to the many senses of the applied nonlinear dynamicist. Sprinkled among a systematic discussion of problem construction, graph representations of delay-coupled problems, and vectorized formulas for problem discretization, the paper includes an original derivation using Lagrangian sensitivity analysis of phase response functionals for periodic-orbit problems in abstract Banach spaces, as well as a demonstration of the regularizing benefits of multi-dimensional manifold continuation for near-singular problems analyzed using real-time experimental data.

This work aims to propose and design a class of networks of coupled linear and nonlinear oscillators, in which short bursts of exogenous excitation result in sustained endogenous network activity that returns to a quiescent state only after a characteristic time and along a different path than when originally excited. The desired hysteretic behavior is obtained through the coupling of self-excited oscillations with purposely designed rate laws for slowly-varying nodal parameters, governed only by local interactions in the network. The proposed architecture and the sought dynamics take inspiration from complex biological systems that combine endogenous energy sources with a paradigm for distributed sensing and information processing. In this paper, the network design problem considers arbitrary topologies and investigates the dependence of the desired response on model parameters, as well as on the placement of a single nonlinear node in an otherwise linear network. Perturbation analysis in various asymptotic parameter limits is used to define the proposed internal dynamics. Parameter continuation techniques validate the asymptotic results numerically and demonstrate their robustness over finite ranges of parameter values. Both approaches suggest a nontrivial dependence of the optimal distribution of nonlinearity on the network topology.

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package coco demonstrates successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.

In this paper, we discuss an original covariance boundary-value problem that captures the effects of white noise on the local behavior near stable periodic orbits of the corresponding deterministic dynamical systems. The methodology relies on suitably constructed adjoint variables to project the dynamics onto locally transversal hyperplanes and ensure the existence of a unique solution. For each such hyperplane, the computed co-variance matrix describes an approximately Gaussian, stationary distribution that highlights directions of particular sensitivity to noise. In contrast to previous formulations, the boundary-value problem analyzed in this paper makes predictions in the original state-space variables rather than in terms of a reduced set of local coordinates for hyperplanes perpendicular to the local vector field. The formulation is compatible with the general form of an augmented continuation problem for the software package COCO; a non-adaptive version has been implemented as a general-purpose constructor for the periodic-orbit toolbox included in the COCO release. We illustrate the efficacy of the proposed formulation and toolbox by analyzing noise-induced behavior near limit cycles in two examples of nonlinear dynamical systems with autonomous drift terms. The analysis presented in this study can be used to design safe operating regimes for systems working in stochastic environments, as well as to design optimum working conditions for systems utilizing noise, such as energy-harvesting applications.

This paper investigates the near-resonance response to exogenous excitation of a class of networks of coupled linear and nonlinear oscillators with emphasis on the dependence on network topology, distribution of nonlinearities, and damping ratios. The analysis shows a qualitative transition between the behaviors associated with the extreme cases of all linear and all nonlinear oscillators, respectively, even allowing for such a transition under continuous variations in the damping ratios but for fixed topology. Theoretical predictions for arbitrary members of the network class using the multiple-scales perturbation method are validated against numerical results obtained using parameter continuation techniques. The latter include the tracking of families of quasi-periodic invariant tori emanating from saddle-node and Hopf bifurcations of periodic orbits. In networks in the class of interest with special topology, 1:1 and 1:3 internal resonances couple modes of oscillation, and the conditions to suppress the influence of these resonances are explored.

We consider optimal control problems on networks with input homogeneity and show a reduction to a universal set of reference problems without differential constraints, whose stationary points may be found analytically. Under suitable nondegeneracy conditions, the derivation shows that input homogeneity results in constant candidate optimal control inputs for the problems of maximizing the terminal response, minimizing the control effort, or minimizing the terminal time, and that these control inputs depend smoothly on the problem data. These predictions are validated using numerical analysis of problems of synchronization of coupled phase oscillators and spreading dynamics on time-varying networks.

This paper proposes two novel adaptive control designs for the feedback signals used in the control-based continuation paradigm to track families of periodic orbits of periodically excited dynamical systems, including black box simulation models and physical experiments. The proposed control designs rely on modifications to the classical model reference adaptive control framework and the more recent $${\mathscr {L}}_1$$ adaptive control architecture, in which an additional low-pass filter is used to ensure guaranteed transient performance and robustness to time delays in the control input even in the limit of arbitrarily large adaptive gains. In contrast to the proportional control formulations that have been used in the literature on control-based continuation, the proposed control designs achieve stable performance with a minimum of parameter tuning. In the context of a class of linear systems with matched uncertainties, the paper demonstrates the successful integration of adaptive control feedback in control-based continuation. Specifically, the control designs are shown to ensure that the control input stabilizes the sought periodic orbits of the uncontrolled system and vanishes along these orbits, provided that an a priori unknown reference input is chosen appropriately. Numerical results obtained using the coco software package demonstrate how the combination of a nonlinear solver (Newton’s method) with the pseudo-arclength parameter continuation scheme can be used to trace the correct choice for the reference input under variations in an excitation parameter.

We create a design tool that allows for modeling the behavior of soft robotic manipulators consisting of serially concatenated, fiber-reinforced elastomeric enclosures (FREEs) as they are pressurized. This tool uses numerical continuation to solve the governing multisegment boundary-value problem as the actuation pressure is varied, accounting for self-contacts via imposition of constraint normal forces. The methodology highlights limitations of existing design techniques for multisegment FREEs that only consider the configuration at the final pressure; achieving complicated tasks such as self-knotting may require considering self-contact in the design process. It is also shown that multiple equilibrium shapes of a multisegment FREE can co-exist across ranges of the actuation pressure. At the extremes of such ranges, we anticipate that small changes in the actuation pressure will result in large changes to the manipulator configuration during pressurization.

This article proposes a methodology for integrating adaptive control with the control-based continuation paradigm for a class of uncertain, linear, discrete-time systems. The proposed adaptive control strategies aim to stabilize the closed-loop dynamics with convergence toward a known reference input, such that the dynamics approach the open-loop fixed point if the reference input is chosen to make the steady-state control input equal 0. This enables the tracking of a parameterized branch of open-loop fixed points using methods of numerical continuation without specific knowledge about the system. We implement two different adaptive control strategies: model-reference adaptive control and pole-placement adaptive control. Both implementations achieve the desired objectives for the closed-loop dynamics and support parameter continuation. These properties, as well as the boundedness of system states and control inputs, are guaranteed provided that certain stability conditions are satisfied. Besides, the tuning effort is significantly reduced in the adaptive control schemes compared with traditional proportional–derivative controllers and linear state-space feedback controllers.

This paper generalizes a previously-conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package coco on periodic orbits of both linear and nonlinear delay-differential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs.

We generalize the successive continuation paradigm introduced by Kernévez and Doedel [1] for locating locally optimal solutions of constrained optimization problems to the case of simultaneous equality and inequality constraints. The analysis shows that potential optima may be found at the end of a sequence of easily-initialized separate stages of continuation, without the need to seed the first stage of continuation with nonzero values for the corresponding Lagrange multipliers. A key enabler of the proposed generalization is the use of complementarity functions to define relaxed complementary conditions, followed by the use of continuation to arrive at the limit required by the Karush-Kuhn-Tucker theory. As a result, a successful search for optima is found to be possible also from an infeasible initial solution guess. The discussion shows that the proposed paradigm is compatible with the staged construction approach of the coco software package. This is evidenced by a modified form of the coco core used to produce the numerical results reported here. These illustrate the efficacy of the continuation approach in locating optimal solutions of an objective function along families of two-point boundary value problems and in optimal control problems.

SUMMARYThis paper reports on laboratory and field experimental results for controlled robotic manipulators operating on moving platforms with unmodeled dynamics. The aim is to validate theoretical predictions for the dependence on control parameters of an adaptive control strategy. In addition, the results provide insight into different discretizations of the continuous-time formulation, suggesting the most suitable discretization scheme for hardware implementation. The second set of experimental results, obtained from an implementation of the control framework for synchronization and consensus in networks of robotic manipulators, similarly validate theoretical predictions on the sensitivity to network communication delays.