Continuous-time Solution Research Articles

Analysis of stochastic SEIR(S) models with random total populations and variable diffusion rates

A stochastic SEIR(S) model with random total population, overall saturation constant K > 0 and general, local Lipschitz-continuous diffusion rates is presented. We prove the existence of unique, Markovian, continuous time solutions w.r.t. filtered, complete probability spaces on certain, bounded 4D prisms. The total population N ( t ) is governed by kind of stochastic logistic equations, which allows to have an asymptotically stable maximum population constant K > 0 . Under natural conditions on our SEIR(S) model, we establish asymptotic stochastic and moment stability of the disease-free and endemic equilibria. Those conditions naturally depend on the basic reproduction number R 0 , the growth parameter μ > 0 and environmental noise intensity σ 5 2 coupled with the maximum threshold K 2 of total population N ( t ) . For the mathematical proofs, the technique of appropriate Lyapunov functionals V ( S ( t ) , E ( t ) , I ( t ) , R ( t ) ) is exploited. Some numerical simulations of the expected Lyapunov functionals E [ V ( S , E , I , R ) ] depending on several parameters and time $t$ support our findings.

Graph Approximating SM Discretization

Modern practical sliding mode (SM) control (SMC) involves discrete noisy sampling. Standard SMC discretizations feature the chattering effect, while implicit Euler methods need significant model knowledge, constant sampling steps and are difficult in application for higher relative degrees and multiple discontinuities. The proposed universal discretization method is based on the graph approximation of Filippov inclusions. It guaranties convergence to the continuous-time solutions and is computationally-simple. For the first time we demonstrate low-chattering discretization in the presence of large noises and multiple discontinuities.

Zero-order hold discretization of general state space systems with input delay

Abstract Plethora of applications in physics and engineering are dealing with systems that are subject to input delays. Despite the scientific focus, previous research investigated only state space systems with input delays. Here, we investigate the discretization of generalized state space system with input delay by using the zero-order hold method. Firstly, the solution of a continuous time, generalized state space system with input delay is addressed. By applying the appropriate zero-order hold sampling method, we transform a continuous time linear system into the equivalent discrete time system, while the discrete time solution of the equivalent system is analytically presented. Finally, we use local error metric to estimate the difference between the continuous time and discrete time solutions.

Dynamic discretization discovery for solving the Continuous Time Inventory Routing Problem with Out-and-Back Routes

In time dependent models, the objective is to find the optimal times (continuous) at which activities occur and resources are utilized. These models arise whenever a schedule of activities needs to be constructed. A common approach consists of discretizing the planning time and then restricting the decisions to those time points. However, this approach leads to very large formulations that are intractable in practice. In this work, we study the Continuous Time Inventory Routing Problem with Out-and-Back Routes (CIRP-OB). In this problem, a company manages the inventory of its customers, resupplying a single product from a single facility during a finite time horizon. The product is consumed at a constant rate (product per unit of time) by each customer. The customers have local storage capacity. The goal is to find the minimum cost delivery plan with out-and-back routes only that ensures that none of the customers run out of product during the planning period. We study the Dynamic Discovery Discretization algorithm (DDD) to solve the CIRP-OB by using partially constructed time-expanded networks. This method iteratively discovers time points needed in the network to find optimal continuous time solutions. We test this method by using randomly generated instances in which we find provable optimal solutions.

Learning generalized Nash equilibria in multi-agent dynamical systems via extremum seeking control

In this paper, we consider the problem of learning a generalized Nash equilibrium (GNE) in strongly monotone games. First, we propose semi-decentralized and distributed continuous-time solution algorithms that use regular projections and first-order information to compute a GNE with and without a central coordinator. As the second main contribution, we design a data-driven variant of the former semi-decentralized algorithm where each agent estimates their individual pseudogradient via zeroth-order information, namely, measurements of their individual cost function values, as typical of extremum seeking control. Third, we generalize our setup and results for multi-agent systems with nonlinear dynamics. Finally, we apply our methods to connectivity control in robotic sensor networks and almost-decentralized wind farm optimization.

A Fast and Fully Parallel Analog CMOS Solver for Nonlinear PDEs

A general-purpose analog computing method is proposed to compute the continuous-time solutions of nonlinear partial differential equations (PDEs). The discrete-time difference operator in the standard finite difference time domain (FDTD) method is replaced by continuous-time delay operators that can be realized using analog all-pass filters. The resulting spatially discrete time-continuous (SDTC) update equations are realized using analog circuits which compute continuous-time solutions of the PDE with prescribed initial and boundary conditions. The proposed concept is demonstrated in simulation via an integrated circuit (IC) design of a nonlinear acoustic wave equation solver in 180 nm CMOS technology. Analog arithmetic operations (multiply, scale, and add) are realized in parallel using fully differential op-amps and analog multipliers. The proposed IC computes the PDE solution in parallel at 33 discrete spatial points and has a simulated bandwidth and power consumption of approximately 2 MHz and 3 W, respectively. The performance of the IC is simulated using foundry-supplied device models and quantified using i) the mean squared difference between the circuit simulation results and FDTD simulations, and ii) the noise to signal energy ratio. Acceptable accuracy is obtained, with error metric values varying between -7 and -30 dB for various configurations of the problem. Comparison of the custom analog IC simulations with MATLAB- and C-based FDTD code running on a modern workstation shows an expected average speedup of 205× and 140×, respectively.

Properties and computation of continuous-time solutions to linear systems

We investigate solutions to the system of linear equations (SoLE) in both the time-varying and time-invariant cases, using both gradient neural network (GNN) and Zhang neural network (ZNN) designs. Two major limitations should be overcome. The first limitation is the inapplicability of GNN models in time-varying environment, while the second constraint is the possibility of using the ZNN design only under the presence of invertible coefficient matrix. In this paper, by overcoming the possible limitations, we suggest, in all possible cases, a suitable solution for a consistent or inconsistent linear system. Convergence properties are investigated as well as exact solutions.

Approximate Transverse Feedback Linearization Under Digital Control

Thanks to a suitable redesign of the maps involved in the continuous-time solution, a digital design procedure preserving transverse feedback linearization up to a prefixed order of approximation in the sampling period is described. Simulated examples illustrate the results.

A Semi-global Hybrid Sensorless Observer for Permanent Magnet Synchronous Machines with Unknown Mechanical Model

In this paper, we present a hybrid sensorless observer for Permanent Magnet Synchronous Machines, with no a priori knowledge of the mechanical dynamics and without the typical assumption of constant or slowly-varying speed. Instead, we impose the rotor speed to have a constant (unknown) sign and a non-zero magnitude at all times. For the design of the proposed scheme, we adopt meaningful Lie group formalism to describe the rotor position as an element of the unit circle. This choice, however, leads to a non-contractible state space, and therefore it introduces topological constraints that complicate the achievement of global/semi-global and robust results. In this respect, we show that the proposed observer, which augments a recent continuous-time solution, achieves semi-global practical asymptotic stability by periodically resetting the estimates. As highlighted in the simulation results, the novel hybrid strategy leads to improved transient performance, notably without any modification of the gains employed in the continuous-time version. These features motivate to augment the observer with a discrete-time identifier, leading to significantly faster rotor flux reconstruction.

Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2017), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.

Continuous-Time Algorithms for Solving Maxwell’s Equations Using Analog Circuits

Two analog computing methods are proposed to compute the continuous-time solutions of one-dimensional (1-D) Maxwell’s equations. In the first method, the spatial domain partial derivatives in the governing partial differential equation (PDE) are approximated using discrete finite differences while applying the Laplace transformation along the time dimension. The resulting spatially discrete time-continuous update equation is utilized to design an analog circuit that can compute the continuous-time solution. The second method replaces the discrete-time difference operators in the standard finite difference time domain (FDTD) cell (Yee cell) using continuous-time delay operators, which can be realized using analog all-pass filters. Both methods have been simulated using ideal analog circuits in Cadence Spectre for the Dirichlet, Neumann, and radiation boundary conditions. The performance of the proposed methods has been quantified using: i) mean squared differences between the results and fully discrete FDTD simulations and ii) the noise to signal energy ratio. Both methods have been extended to design the analog circuits that compute the continuous-time solution of the 1-D and 2-D wave equations. The 1-D wave equation solver is simulated with a dominant-pole model (which better approximates the non-ideal circuit behavior) along with a propagation delay compensation technique. The experimental results from a simplified board-level low-frequency implementation are also presented. The key challenges toward CMOS implementations of the proposed solvers are identified and briefly discussed with possible solutions.

Log-periodicity in piecewise ballistic superdiffusion: Exact results

Small log-periodic oscillations have been observed in many systems and have previously been studied via renormalization-group approaches in the context of critical phenomena [Gluzman and Sornette, Phys. Rev. E 65, 036142 (2002); Derrida and Giacomin, J. Stat. Phys. 154, 286 (2014)]. Here we report their appearance in a random walk model with damaged memory, and we develop an exact discrete-time solution, free from adjustable parameters. Our results shed light on log-periodicity and how it arises. We also discuss continuous-time approaches to the solution along with their limitations and advantages. We show that, as a direct consequence of memory damage, the first moment for the model acquires piecewise ballistic behavior. The piecewise segments are separated by regularly placed singular points. Log-periodicity in this model is seen to be due to memory damage. Remarkably, piecewise ballistic behavior is only observed if one uses the discrete-time solution, because the continuous-time solution does not correctly account for the model's discrete-time dynamics.

Continuous-time mean variance portfolio with transaction costs: a proximal approach involving time penalization

ABSTRACTThis paper proposes a new continuous-time optimization solution that enables the computation of the portfolio problem (based on the utility option pricing and the shortfall risk minimization). We first propose a dynamical stock price process, and then, we transform the solution to a continuous-time discrete-state Markov decision processes. The market behavior is characterized by considering arbitrage-free and assessing transaction costs. To solve the problem, we present a proximal optimization approach, which considers time penalization in the transaction costs and the utility. In order to include the restrictions of the market, as well as those that imposed by the continuous-time space, we employ the Lagrange multipliers approach. As a result, we obtain two different equations: one for computing the portfolio strategies and the other for computing the Lagrange multipliers. Each equation in the portfolio is an optimization problem, for which the necessary condition of a maximum/minimum is solved employing the gradient method approach. At each step of the iterative proximal method, the functional increases and finally converges to a final portfolio. We show the convergence of the method. A numerical example showing the effectiveness of the proposed approach is also developed and presented.

Discrete-time theorems for global and pointwise dichotomies of cocycles over semiflows

In this paper, we consider linear skew-product semiflows on bundles of Banach fibers over a locally compact metric space. Our aim is to give discrete-time theorems for the existence of global and pointwise continuous-time dichotomies with no invariant unstable manifolds. We involve here a concept of exponential dichotomy for skew-product semiflows weaker than the concept used by Sacker and Sell (J Differ Equ 15:429–458, 1974; 22:478–522, 1976) and Magalhaes (in: Chow, Hale (eds) Dynamics of infinite dimensional systems, NATO Advanced Science Institutes Series F: Computer and Systems Sciences, Springer, New York, vol 37, pp 161–168, 1987); our definition (of no past exponential dichotomy) follows roughly the definition given by Chow and Leiva (Proc Am Math Soc 124:1071–1081, 1996) in the sense that we allow the unstable subspace to have infinite dimension. The main improvement is that we go even more general and we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable subspace is invariant under the cocycle). Roughly speaking, we prove that if the solution of the corresponding inhomogeneous variational difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the variational homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the above condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p-summable spaces, sequence Orlicz spaces, etc.). Since we use a discrete-time technique we are not forced to require any continuity or measurability hypotheses on the trajectories of the exponentially bounded cocycle. Also, it is worth to mention that from discrete-time conditions we get informations about the continuous-time behavior of the solutions of differential variational equations.

A Consumption Based Term Structure Model w ith Habit Utility

This paper proposes a term structure of interest rates model that modifies and extends the Campbell and Cochrane (1999) surplus consumption framework. The distinguishing contributions are tractable, continuous-time analytical solutions for the term structure of interest rate generating a realistic upward sloping yield curve. Despite the focus on the term structure, the model matches plausible equity quantities. For the interest rate, the model is able to account for the moments of bond yields at numerous maturities and produce countercyclical bond risk premia as seen in the data. Moreover, the model captures reasonable time series fluctuation on real interest rates. However, the model has difficulties reproducing empirical deviations from the expectations hypothesis.

Continuous-time general link transmission model with simplified fanning, Part I: Theory and link model formulation

The kinematic wave theory is widely used to simulate traffic flows on road segments. Link transmission models are methods to find a solution to the kinematic wave model, however, their computational efficiency heavily relies on the shape of the fundamental diagram that is used as input. Despite the limitations and drawbacks of triangular and piecewise linear fundamental diagrams, they remain popular because they result in highly efficient algorithms. Using smooth nonlinear branches is often preferred in terms of realism and other desirable properties, but this comes at a significantly higher computational cost and requires time discretization to find an approximate solution. In this paper we consider a nonlinear fundamental diagram as input and propose on-the-fly multi-step linearization techniques to simplify expansion fans. This leads to two simplified link transmission models that can be solved exactly in continuous time under the assumption of piecewise stationary travel demand. One of the models simplifies to shockwave theory in case of a single step. We show that embedding shockwave theory in the link transmission model allows for finding an exact solution in continuous time and we discuss the potential for the design of efficient event-based algorithms for general networks.

The Continuous-Time Service Network Design Problem

Consolidation carriers transport shipments that are small relative to trailer capacity. To be cost effective, the carrier must consolidate shipments, which requires coordinating their paths in both space and time; i.e., the carrier must solve a service network design problem. Most service network design models rely on discretization of time—i.e., instead of determining the exact time at which a dispatch should occur, the model determines a time interval during which a dispatch should occur. While the use of time discretization is widespread in service network design models, a fundamental question related to its use has never been answered: Is it possible to produce an optimal continuous-time solution without explicitly modeling each point in time? We answer this question in the affirmative. We develop an iterative refinement algorithm using partially time-expanded networks that solves continuous-time service network design problems. An extensive computational study demonstrates that the algorithm not only is of theoretical interest but also performs well in practice.

The Blessed Virgin and the Two Time-Series: Hervaeus Natalis and Durand of St. Pourçain on Limit Decision

This paper examines the accounts of limit decision advanced by Hervaeus Natalis and Durand of St. Pourçain in their respective discussions of the sanctification of the Blessed Virgin. Hervaeus and Durand argue, against Aristotle, that the temporal limits of certain changes, including Mary’s sanctification, should be assigned in discrete rather than continuous time. The paper first considers Hervaeus’ discussion of limit decision and argues that, for Hervaeus, a solution of temporal limits in terms of discrete time can coexist with an Aristotelian continuous time solution because Hervaeus takes continuous and discrete time to be two non-intersecting, but correlated time-series. The paper next examines Durand’s account of limit decision and argues that Durand rejects Hervaeus’ correlation assumption as well as Aristotle’s continuous time solution.

Active Fault Isolation: A Duality-Based Approach via Convex Programming

This paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous-time linear systems. Given a set of faults known a priori, the system is modeled by a finite family of linear time-invariant systems, accounting for one healthy and several faulty configurations. By assuming bounded disturbances and using a residual generator, an invariant set and its projection in the residual space (i.e., its limit set) are computed for each system configuration. Each limit set, related to a single system configuration, is parameterized with respect to the system input. Thanks to this design, active fault isolation can be guaranteed by the computation of a test input, either constant or periodic, such that the limit sets associated with different system configurations are separated, and the residual converges toward one limit set only. In order to alleviate the complexity of the explicit computation of the limit set, an implicit dual representation is adopted, leading to efficient procedures, based on quadratic programming , for computing the test input. The developed methodology offers a competent continuous-time solution to the optimization-based computation of the test input via Hahn–Banach duality. Simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution, even in relatively large state-dimensional problems.

Portfolio Selection with Transaction Costs and Jump-Diffusion Asset Dynamics I: A Numerical Solution

We derive allocation rules under isoelastic utility for a mixed jump-diffusion process in a two-asset portfolio selection problem with finite horizon in the presence of proportional transaction costs. We adopt a discrete time formulation, let the number of periods go to infinity, and show that it converges efficiently to the continuous time solution for the cases where this solution is known. We then apply this discretization to derive numerically the boundaries of the region of no transactions. Our discrete-time numerical approach outperforms alternative continuous-time approximations of the problem.