The deterministic analog of the Markov property of a time-homogeneous Markov process is the semig... more The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N. V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.
data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modificatio... more data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement s > s crit is necessary.
* The first author was partially supported by NSF grants DMS-0204651 and DMS-0604994. † The secon... more * The first author was partially supported by NSF grants DMS-0204651 and DMS-0604994. † The second author was partially supported by NSF grants DMS-0200670 and DMS-0436403. Appendix 1: The W 1,1 image of a connected set 53 References 54 2 Review and examples This section contains a brief review of background material and several examples. 2.1 Homotopy classification of S 2-valued maps The homotopy classification of maps M → S 2 has been known since the nineteen-thirties. First Hopf [30] classified maps from S 3 into S 2 and defined a complete integer-valued invariant that counts the linking number of the inverse image of a pair of regular values. In
A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusi... more A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semiprocesses in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.
The phenomenon of the global (in time) dispersive smoothing for the "free" Schrödinger evolution ... more The phenomenon of the global (in time) dispersive smoothing for the "free" Schrödinger evolution can be described as follows: For any distribution f of compact support, the solution ψ(t, x) of the Cauchy problem (1 i ∂ ∂t − ∆) ψ(t, x) = 0, t > 0, ψ(0, x) = f (x), x ∈ R n , is infinitely differentiable with respect to t and x, when t > 0 and x ∈ R n. This is equivalent to saying that the corresponding fundamental solution (= the solution S 0 (t, x, y) of the initial value problem with f (x) = δ(x − y)) is infinitely differentiable with respect to t, x and y, when t > 0. And we have, indeed, S 0 (t, x, y) = e −in π 4 (4πt) − n 2 exp{i |x − y| 2 /4t}, with the only singularity at t = 0. One would expect that dispersive smoothing should survive "small" perturbations of the free Hamiltonian H 0 = −∆. The problem is to determine what perturbations are "small". The case when the perturbed Hamiltonian has the form H = H 0 + V with a potential V = V (x), has been examined in [Ze], [OF], [Ki], [CFKS]. The dispersive smoothing takes place, for example, if the potential is infinitely differentiable, and it and all its derivatives are bounded, [Ze], [OF]. On the other hand, if V (x) grows quadratically or faster at infinity, then the singularities may resurrect, as the example of the quantum harmonic oscillator and Mehler's formula show ([Ze], [We], [CFKS], [MF]). If the perturbation affects the metric of the space, i.e., if H 0 = −∆ is replaced by H = − n j,k=1 ∂ ∂x j a j,k (x) ∂ ∂x k , the problem apparently becomes more subtle. Practically no information on global dispersive smoothing
Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evol... more Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, or Nature's intervention, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.
data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modificatio... more data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement s > s crit is necessary.
We construct a parametrix for the initial-value problem for the time- dependent Schrodinger equat... more We construct a parametrix for the initial-value problem for the time- dependent Schrodinger equation in Rn with coecients that stabilize at infinity. Our
Difference Equations, Discrete Dynamical Systems and Applications, 2016
We are interested in time evolution of systems that switch their modes of operation at discrete m... more We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system, when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.
We discuss matching control laws for underactuated systems. We previously showed that this class ... more We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely charactarized by a linear system of first order partial differential equations for one set of variables (λ) followed by a linear system of first order PDEs for the second set of variables (g, V). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the λ-equations. We give four examples illustrating different features of matching control laws. The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case. 1. Introduction. Effective procedures for designing control laws are very important in nonlinear control theory. Explicit analytic formulae fo...
The deterministic analog of the Markov property of a timehomogeneous Markov process is the semigr... more The deterministic analog of the Markov property of a timehomogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on the measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N.V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.
Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications, 2020
A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusi... more A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semiprocesses in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.
Nonlinear Dynamics and Renormalization Group, Apr 9, 2001
There are many interesting mathematical problems in control theory. In this paper we will discuss... more There are many interesting mathematical problems in control theory. In this paper we will discuss problems and techniques related to underactuated systems. An underactuated
The deterministic analog of the Markov property of a time-homogeneous Markov process is the semig... more The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N. V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.
data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modificatio... more data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement s > s crit is necessary.
* The first author was partially supported by NSF grants DMS-0204651 and DMS-0604994. † The secon... more * The first author was partially supported by NSF grants DMS-0204651 and DMS-0604994. † The second author was partially supported by NSF grants DMS-0200670 and DMS-0436403. Appendix 1: The W 1,1 image of a connected set 53 References 54 2 Review and examples This section contains a brief review of background material and several examples. 2.1 Homotopy classification of S 2-valued maps The homotopy classification of maps M → S 2 has been known since the nineteen-thirties. First Hopf [30] classified maps from S 3 into S 2 and defined a complete integer-valued invariant that counts the linking number of the inverse image of a pair of regular values. In
A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusi... more A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semiprocesses in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.
The phenomenon of the global (in time) dispersive smoothing for the "free" Schrödinger evolution ... more The phenomenon of the global (in time) dispersive smoothing for the "free" Schrödinger evolution can be described as follows: For any distribution f of compact support, the solution ψ(t, x) of the Cauchy problem (1 i ∂ ∂t − ∆) ψ(t, x) = 0, t > 0, ψ(0, x) = f (x), x ∈ R n , is infinitely differentiable with respect to t and x, when t > 0 and x ∈ R n. This is equivalent to saying that the corresponding fundamental solution (= the solution S 0 (t, x, y) of the initial value problem with f (x) = δ(x − y)) is infinitely differentiable with respect to t, x and y, when t > 0. And we have, indeed, S 0 (t, x, y) = e −in π 4 (4πt) − n 2 exp{i |x − y| 2 /4t}, with the only singularity at t = 0. One would expect that dispersive smoothing should survive "small" perturbations of the free Hamiltonian H 0 = −∆. The problem is to determine what perturbations are "small". The case when the perturbed Hamiltonian has the form H = H 0 + V with a potential V = V (x), has been examined in [Ze], [OF], [Ki], [CFKS]. The dispersive smoothing takes place, for example, if the potential is infinitely differentiable, and it and all its derivatives are bounded, [Ze], [OF]. On the other hand, if V (x) grows quadratically or faster at infinity, then the singularities may resurrect, as the example of the quantum harmonic oscillator and Mehler's formula show ([Ze], [We], [CFKS], [MF]). If the perturbation affects the metric of the space, i.e., if H 0 = −∆ is replaced by H = − n j,k=1 ∂ ∂x j a j,k (x) ∂ ∂x k , the problem apparently becomes more subtle. Practically no information on global dispersive smoothing
Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evol... more Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, or Nature's intervention, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.
data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modificatio... more data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement s > s crit is necessary.
We construct a parametrix for the initial-value problem for the time- dependent Schrodinger equat... more We construct a parametrix for the initial-value problem for the time- dependent Schrodinger equation in Rn with coecients that stabilize at infinity. Our
Difference Equations, Discrete Dynamical Systems and Applications, 2016
We are interested in time evolution of systems that switch their modes of operation at discrete m... more We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system, when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.
We discuss matching control laws for underactuated systems. We previously showed that this class ... more We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely charactarized by a linear system of first order partial differential equations for one set of variables (λ) followed by a linear system of first order PDEs for the second set of variables (g, V). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the λ-equations. We give four examples illustrating different features of matching control laws. The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case. 1. Introduction. Effective procedures for designing control laws are very important in nonlinear control theory. Explicit analytic formulae fo...
The deterministic analog of the Markov property of a timehomogeneous Markov process is the semigr... more The deterministic analog of the Markov property of a timehomogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on the measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N.V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.
Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications, 2020
A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusi... more A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semiprocesses in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.
Nonlinear Dynamics and Renormalization Group, Apr 9, 2001
There are many interesting mathematical problems in control theory. In this paper we will discuss... more There are many interesting mathematical problems in control theory. In this paper we will discuss problems and techniques related to underactuated systems. An underactuated
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