# UMBC Mathematics and Statistics Department

## Permanent URI for this collection

## Browse

### Recent Submissions

Item Semigroup Methods for Poroelastic Multi-Physics Systems Describing Biological Tissues(2024/01/01) Gurvich, Elena; Webster, Justin T; Mathematics and Statistics; Mathematics, AppliedShow more This thesis presents novel work in the mathematical theory of poroelasticity, which was first phenomenologically developed by Biot and Zenisek during the mid-last century. The theory relates saturated porous structural deformations to fluid pressure changes within, and blossomed through many applications in the geosciences (e.g., seismic and petroleum engineering). At the outset of the 21st century, poroelasticity proved to be a revolutionary incorporation to the biological fields (e.g. biomedical engineering, arterial stents, scaffolding), owing to the poroelastic nature of biological tissues. For the parameters of physical interest, a quasi-static approximation induces dynamics, which can be represented as an implicit evolution. Moreover, compressibility in Biot's equations is a significant consideration. In the incompressible limit, Biot’s model degenerates This dissertation will present a biologically motivated multilayered system, composed of the coupled dynamics of a 3D poroelastic structure, a poroelastic plate, and an incompressible free Stokes flow. We propose two constituent sub-problems, to gain a better understanding of this extremely complex system. First, a complete well-posedness analysis of the poroelastic plate is shown utilizing variational tools. Secondly, Biot-Stokes filtration is proposed with Beavers-Joseph-Saffman coupling conditions on a fixed 2D interface. A semigroup approach is used to bypass the issues with mismatched trace regularities on the interface; thus guaranteeing strong and generalized solutions. Then the existence of weak solutions, including the degenerate case, is provided by argument by density. The most interesting cases are singular limits, which lead to the use of the theory of abstract implicit, degenerate evolutions, of which the appendix supplies a brief overview. Thus, this thesis provides a clear elucidation of strong solutions and the construction of weak solutions for inertial linear Biot-Stokes filtration systems and uniquely for a poroelastic plate, as well as their regularity through associated estimates.Show more Item Disciplinary Differences in STEM Faculty and Student Use of Learning Objectives: Implications for Teaching and Learning(Taylor & Francis, 2024-07-22) Leupen, Sarah; Williams, Tory; Hodges, Linda C.; Ott, Laura E.; Anderson, Eric C.; Cui, Lili; Nanes, Kalman M.; Perks, H. Mark; Wagner, Cynthia R.Show more Using learning objectives to guide course design is often considered an educational best practice, but little research exists that explores how students use them over time and across courses. We surveyed students on their use and perceived value of learning objectives as the semester progressed across four science, technology, engineering, and mathematics (STEM) disciplines, examined students’ ability to match exam questions with learning objectives, and analyzed how their course performance related to these qualities. We also gathered instructors’ information on their implementation of learning objectives in these courses. We identified distinct disciplinary differences both in students’ use and perceived benefit of learning objectives and in instructors’ implementation of them. Students in less quantitatively focused courses, i.e., biology and organic chemistry, reported valuing and using learning objectives more than students in more quantitatively focused math and physics courses. Students’ ability to match learning objectives with exam questions, however, positively correlated with exam score and final course grade in all our study courses. Our results have implications for considering disciplinary practices for use of learning objectives as instructors design and implement courses, educational researchers plan studies, and assessment specialists formulate institutional assessment plans.Show more Item Semistability of switched linear systems with applications to distributed sensor networks: A generating function approach(IEEE, 2012-03-01) Shen, Jinglai; Hu, Jianghai; Hui, QingShow more This paper investigates semistability of discrete-time, switched linear systems under both deterministic and random switching policies. The notion of semistability pertains to a continuum of initial state dependent equilibria and has found wide applications such as consensus problems in multi-agent systems. The main contributions of the paper are three folds. First, we show that exponential semistability on a common equilibrium space is equivalent to output exponential stability of the switched linear system with a suitably defined output, under both arbitrary and random switchings. Further, their convergence rates are shown to be identical. Second, it is shown that output stability and its convergence rates can be efficiently characterized via the recently developed generating function approach. Third, we consider algorithm development and analysis of resource allocation schemes for topologically changing, distributed sensor networks. We formulate an iteration process of such an algorithm as a switched linear system, and characterize its convergence using the obtained semistability results.Show more Item Stabilization of Switched Linear Systems Using Continuous Control Input against Known Adversarial Switching(IEEE, 2018-06) Hu, Jianghai; Shen, Jinglai; Lee, DonghwanShow more The problem of designing continuous control input to stabilize switched linear control systems against adversarial switching is studied. It is assumed that the continuous controller has access to the current switching mode and can be of the form of an ensemble of mode-dependent state feedback controllers. The fastest stabilizing rate under the given information structure is proposed as a quantitative metric of the system's stabilizability, and its bounds are derived using seminorms. Computation algorithms for the stabilizing rate are developed and illustrated through examples.Show more Item Convex regression via penalized splines: A complementarity approach(IEEE, 2012-10-01) Shen, Jinglai; Wang, XiaoShow more Estimation of a convex function is an important shape restricted nonparametric inference problem with broad applications. In this paper, penalized splines (or simply P-splines) are exploited for convex estimation. The paper is devoted to developing an asymptotic theory of a class of P-spline convex estimators using complementarity techniques and asymptotic statistics. Due to the convex constraints, the optimality conditions of P-splines are characterized by nons-mooth complementarity conditions. A critical uniform Lipschitz property is established for optimal spline coefficients. This property yields boundary consistency and uniform stochastic boundedness. Using this property, the P-spline estimator is approximated by a two-step estimator based on the corresponding least squares estimator, and its asymptotic behaviors are obtained using asymptotic statistic techniques.Show more Item Stabilizing switched linear systems under adversarial switching(IEEE, 2015-12) Hu, Jianghai; Lee, Dong-Hwan; Shen, JinglaiShow more The problem of stabilizing discrete-time switched linear control systems using continuous input by the user and against adversarial switching by an adversary is studied. It is assumed that the adversary has the advantage in that at each time it knows the user's decision on the continuous control input but not vice versa. Stabilizability conditions and bounds on the fastest stabilizing rates are derived. Examples are given to illustrate the results.Show more Item Mathematical Models for the Triaxial Attitude Control Testbed(Taylor & Francis, 2010-08-09) Cho, Sangbum; Shen, Jinglai; Mcclamroch, N. HarrisShow more The Triaxial Attitude Control Testbed has been developed as part of a research program at the University of Michigan on multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models; these include fan actuators, reaction wheel actuators and proof mass actuators that are fixed to the triaxial base body. The models also allow incorporation of unactuated auxiliary bodies that are constrained to move relative to the triaxial base body. The models expose the dynamic coupling between the rotational motion of the triaxial base body, the relative or shape motion of the unactuated auxiliary degrees of freedom, and dynamics associated with actuation mechanisms. Many different model simplifications and approximations are developed. Control models for the triaxial attitude control testbed are formulated that reflect specific assumptions.Show more Item ROBUST DYNAMIC INVERSION DESIGN OF MANEUVERABLE FLIGHT COMMAND TRACKING SYSTEMS(Beijing University of Aeronautics & Astronautics, 1998-10-25) Jinglai, Shen; Jinyuan, GaoShow more The design of a maneuverable flight command tracking system using nonlinear dynamic inversion is briefly introduced at first. And then a robust controller to improve the robustness of dynamic inversion is designed using the method of μ analysis and synthesis. The detailed design process and results are presented. Simulations show that the robust dynamic inversion controller yields good results.Show more Item Stability and Stabilization of Relative Equilibria of Dumbbell Bodies in Central Gravity(AIAA, 2005-09) Sanyal, Amit K.; Shen, Jinglai; McClamroch, N. Harris; Bloch, Anthony M.Show more A dumbbell-shaped rigid body can be used to represent certain large spacecraft or asteroids with bimodal mass distributions. Such a dumbbell body is modeled as two identical mass particles connected by a rigid, massless link. Equations of motion for the five degrees of freedom of the dumbbell body in a central gravitational field are obtained. The equations of motion characterize three orbit degrees of freedom, two attitude degrees of freedom, and the coupling between them. The system has a continuous symmetry due to a cyclic variable associated with the angle of right ascension of the dumbbell body. Reduction with respect to this symmetry gives a reduced system with four degrees of freedom. Relative equilibria, corresponding to circular orbits, are obtained from these reduced equations of motion; the stability of these relative equilibria is assessed. It is shown that unstable relative equilibria can be stabilized by suitable attitude feedback control of the dumbbell.Show more Item Linear Complementarity Systems: Zeno States(SIAM, 2005-01) Shen, Jinglai; Pang, Jong-ShiShow more A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear time-invariant ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the P-property has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly non-Zeno. While related mode-switch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.Show more Item Lyapunov Stability of Complementarity and Extended Systems(SIAM, 2007-01) Camlibel, M. Kanat; Pang, Jong‐Shi; Shen, JinglaiShow more A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear time-invariant ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the P-property has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly non-Zeno. While related mode-switch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.Show more Item Smoothing splines with varying smoothing parameter(Oxford University Press, 2013-06-08) Wang, Xiao; Du, Pang; Shen, JinglaiShow more This paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain. Two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty. The penalty is a function of the design points in order to accommodate local behaviour of the regression function. We show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator, and that the equivalent kernel is spatially dependent. The equivalent kernels for traditional smoothing splines are a special case of this general solution. With the aid of the Green’s function for a two-point boundary value problem, explicit forms of the asymptotic mean and variance are obtained for any interior point. Thus, the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error. Simulation results and an application illustrate the performance of the proposed estimator.Show more Item Uniform Convergence and Rate Adaptive Estimation of Convex Functions via Constrained Optimization(SIAM, 2013-01) Wang, Xiao; Shen, JinglaiShow more This paper discusses asymptotic analysis and adaptive design of convex estimators over the Hölder class under the sup-norm risk and the pointwise risk using constrained optimization and asymptotic statistical techniques. Specifically, convex B-spline estimators are proposed to achieve uniform optimal convergence rates and adaptive procedures. The presence of the convex shape constraint complicates asymptotic performance analysis, particularly uniform convergence analysis. This in turn requires deep understanding of a family of size varying constrained optimization problems on spline coefficients. To address these issues, we establish the uniform Lipschitz property of optimal spline coefficients in the ℓ∞-norm by exploiting the structure of underlying constrained optimization problems. By using this property, polyhedral theory, and statistical techniques, we show that the convex B-spline estimator attains uniform consistency and optimal rates of convergence on the entire interval of interest over the Hölder class under the sup-norm risk and the pointwise risk. In addition, adaptive estimates are constructed under both risks when the Hölder exponent is between one and two. These estimates achieve a maximal risk within a constant factor of the minimax risk over the Hölder class.Show more Item Semistability of switched linear systems with application to PageRank algorithms(Elsevier, 2014-05-01) Shen, Jinglai; Hu, Jianghai; Hui, QingShow more This paper investigates semistability and its computation for discrete-time, switched linear systems under both deterministic and random switching policies. The notion of semistability pertains to a continuum of initial state dependent equilibria, and finds wide applications in multi-agent and distributed network systems. It is shown in this paper that exponential semistability on a common equilibrium space is equivalent to output exponential stability of a reduced switched linear system with a suitably defined output, under arbitrary and random switchings. Besides, their convergence rates are shown to be identical. A generating function based approach is proposed to compute convergence rates of the reduced switched systems under these switching rules. The obtained semistability results are applied to performance analysis of PageRank algorithms for distributed web-page systems subject to topology switching. The iteration processes of these algorithms are formulated as switched linear systems. Their equilibrium properties are studied, and convergence rates are characterized via the semistability techniques and the generating function approach.Show more Item Robust Non-Zenoness of Piecewise Affine Systems with Applications to Linear Complementarity Systems(SIAM, 2014-01) Shen, JinglaiShow more A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear time-invariant ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the P-property has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly non-Zeno. While related mode-switch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.Show more Item Shape restricted smoothing splines via constrained optimal control and nonsmooth Newton’s methods(Elsevier, 2015-03-01) Shen, Jinglai; Lebair, Teresa M.Show more Shape restricted smoothing splines receive considerable attention, motivated by many important applications in science and engineering. In this paper, we consider smoothing splines subject to general linear dynamics and control constraints, and formulate them as finite-horizon constrained linear optimal control problems with unknown initial state and control. By exploring techniques from functional and variational analyses, optimality conditions are developed in terms of variational inequalities. Due to the control constraints, the optimality conditions give rise to a nonsmooth B-differentiable equation of an optimal initial condition, whose unique solution completely determines the shape restricted smoothing spline. A modified nonsmooth Newton’s algorithm with line search is used to solve this equation; detailed convergence analysis of the proposed algorithm is presented. Using techniques from nonsmooth analysis and polyhedral theory, we show the global convergence of the algorithm for shape restricted smoothing splines subject to general polyhedral control constraints.Show more Item Generalized Input-to-State l2-Gains of Discrete-Time Switched Linear Control Systems(SIAM, 2016-01) Hu, Jianghai; Shen, Jinglai; Putta, VamsiShow more A generalized notion of input-to-state ℓ₂-gain is proposed for discrete-time switched linear control systems (SLCSs). Dependent on a discount factor of subsystem matrices, this generalized ℓ₂-gain provides new insight into input-to-state behaviors of the SLCSs under parameter variations. After establishing analytical properties of the generalized ℓ₂-gain, this paper focuses on the generating function approach to the study of the generalized ℓ₂-gain. Important properties of generating functions are derived, and it is shown that their radii of convergence characterize the generalized ℓ₂-gain. Furthermore, iterative algorithms are developed for computing the generating functions with proven uniform or exponential convergence. Numerical results show that these algorithms yield efficient estimates of both the generalized and classical ℓ₂-gains.Show more Item Minimax Lower Bound and Optimal Estimation of Convex Functions in the Sup-Norm(IEEE, 2017-07) Lebair, Teresa M.; Shen, Jinglai; Wang, XiaoShow more Estimation of convex functions finds broad applications in science and engineering; however, the convex shape constraint complicates the asymptotic performance analysis of such estimators. This technical note is devoted to the minimax optimal estimation of univariate convex functions in a given Hölder class. Particularly, a minimax lower bound in the supremum norm (or simply sup-norm) is established by constructing a novel family of piecewise quadratic convex functions in the Hölder class. This result, along with a recent result on the minimax upper bound, gives rise to the optimal rate of convergence for the minimax sup-norm risk of convex functions with the Hölder order between one and two. The present technical note provides the first rigorous justification of the optimal minimax risk for convex estimation on the entire interval of interest in the sup-norm.Show more Item Resilient Stabilization of Switched Linear Control Systems Against Adversarial Switching(IEEE, 2017-01-10) Hu, Jianghai; Shen, Jinglai; Lee, DonghwanShow more This paper studies the problem of stabilizing discrete-time switched linear control systems (SLCSs) using continuous input by a user against adversarial switching by an adversary. It is assumed that at each time the adversary knows the user's decision on the continuous input but not vice versa. A quantitative metric of stabilizability is proposed. Systems at the margin of stabilizability are further classified and studied via the notions of defectiveness and reducibility. Analytical bounds on the stabilizability metric are derived using (semi)norms, with tight bounds provided by extremal norms. Numerical algorithms are also developed for computing this metric. An application example in networked control systems is presented.Show more Item Conewise Linear Systems: Non?Zenoness and Observability(SIAM, 2006-01) Camlibel, M. Kanat; Pang, Jong?Shi; Shen, JinglaiShow more Show more