52 research outputs found
Exact and optimal quadratization of nonlinear finite-dimensional non-autonomous dynamical systems
Quadratization of polynomial and nonpolynomial systems of ordinary
differential equations is advantageous in a variety of disciplines, such as
systems theory, fluid mechanics, chemical reaction modeling and mathematical
analysis. A quadratization reveals new variables and structures of a model,
which may be easier to analyze, simulate, control, and provides a convenient
parametrization for learning. This paper presents novel theory, algorithms and
software capabilities for quadratization of non-autonomous ODEs. We provide
existence results, depending on the regularity of the input function, for cases
when a quadratic-bilinear system can be obtained through quadratization. We
further develop existence results and an algorithm that generalizes the process
of quadratization for systems with arbitrary dimension that retain the
nonlinear structure when the dimension grows. For such systems, we provide
dimension-agnostic quadratization. An example is semi-discretized PDEs, where
the nonlinear terms remain symbolically identical when the discretization size
increases. As an important aspect for practical adoption of this research, we
extended the capabilities of the QBee software towards both non-autonomous
systems of ODEs and ODEs with arbitrary dimension. We present several examples
of ODEs that were previously reported in the literature, and where our new
algorithms find quadratized ODE systems with lower dimension than the
previously reported lifting transformations. We further highlight an important
area of quadratization: reduced-order model learning. This area can benefit
significantly from working in the optimal lifting variables, where quadratic
models provide a direct parametrization of the model that also avoids
additional hyperreduction for the nonlinear terms. A solar wind example
highlights these advantages
Random Finite Set Theory and Optimal Control of Large Collaborative Swarms
Controlling large swarms of robotic agents has many challenges including, but
not limited to, computational complexity due to the number of agents,
uncertainty in the functionality of each agent in the swarm, and uncertainty in
the swarm's configuration. This work generalizes the swarm state using Random
Finite Set (RFS) theory and solves the control problem using Model Predictive
Control (MPC) to overcome the aforementioned challenges. Computationally
efficient solutions are obtained via the Iterative Linear Quadratic Regulator
(ILQR). Information divergence is used to define the distance between the swarm
RFS and the desired swarm configuration. Then, a stochastic optimal control
problem is formulated using a modified L2^2 distance. Simulation results using
MPC and ILQR show that swarm intensities converge to a target destination, and
the RFS control formulation can vary in the number of target destinations. ILQR
also provides a more computationally efficient solution to the RFS swarm
problem when compared to the MPC solution. Lastly, the RFS control solution is
applied to a spacecraft relative motion problem showing the viability for this
real-world scenario.Comment: arXiv admin note: text overlap with arXiv:1801.0731
Unconditionally energy stable fully discrete schemes for a chemo-repulsion model
This work is devoted to studying unconditionally energy stable
and mass-conservative numerical schemes for the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, such that
∂tu − Δu −∇· (u∇v) = 0 in Ω, t> 0,
∂tv − Δv + v = u in Ω, t> 0,
in a bounded domain Ω ⊆ Rd, d = 2, 3. By using a regularization technique,
we propose three fully discrete Finite Element (FE) approximations. The
first one is a nonlinear approximation in the variables (u, v); the second one
is another nonlinear approximation obtained by introducing σ = ∇v as an
auxiliary variable; and the third one is a linear approximation constructed by
mixing the regularization procedure with the energy quadratization technique, in which other auxiliary variables are introduced. In addition, we study the well-posedness of the numerical schemes, proving unconditional existence of solution, but conditional uniqueness (for the nonlinear schemes). Finally, we compare the behavior of such schemes throughout several numerical simulations and provide some conclusions.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Vicerrectoría de Investigación y Extensión (Universidad Industrial de Santander
Mathematical Modeling of Biological Systems
Mathematical modeling is a powerful approach supporting the investigation of open problems in natural sciences, in particular physics, biology and medicine. Applied mathematics allows to translate the available information about real-world phenomena into mathematical objects and concepts. Mathematical models are useful descriptive tools that allow to gather the salient aspects of complex biological systems along with their fundamental governing laws, by elucidating the system behavior in time and space, also evidencing symmetry, or symmetry breaking, in geometry and morphology. Additionally, mathematical models are useful predictive tools able to reliably forecast the future system evolution or its response to specific inputs. More importantly, concerning biomedical systems, such models can even become prescriptive tools, allowing effective, sometimes optimal, intervention strategies for the treatment and control of pathological states to be planned. The application of mathematical physics, nonlinear analysis, systems and control theory to the study of biological and medical systems results in the formulation of new challenging problems for the scientific community. This Special Issue includes innovative contributions of experienced researchers in the field of mathematical modelling applied to biology and medicine
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