Partial Stabilization of Input-Output Contact Systems on a Legendre Submanifold

“…Let us start by reviewing briefly (and extend in its scope) the lift of controlled dynamical systems proposed recently in [3] (see also [6][7][8][9][10][11][12]). We consider a dynamical system of interest, represented by the equationsẋ…”

Section: About the Lift Of Controlled Systemsmentioning

confidence: 99%

“…In particular, in [3] the concepts of "conservative contact systems", and "conservative control contact systems" have been introduced. Moreover, in [6][7][8][9][10][11][12] the properties of controllability and stability of the flow have been thoroughly analyzed.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics and evolutionary biology through optimal control

Bravetti

Padilla

2019

Automatica

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We consider a particular instance of the lift of controlled systems recently proposed in the theory of irreversible thermodynamics and show that it leads to a variational principle for an optimal control in the sense of Pontryagin. Then we focus on two important applications: in thermodynamics and in evolutionary biology. In the thermodynamic context, we show that this principle provides a dynamical implementation of the Second Law, which stabilizes the equilibrium manifold of a system. In the evolutionary context, we discuss several interesting features: it provides a robust scheme for the coevolution of the population and its fitness landscape; it has a clear informational interpretation; it recovers Price equation naturally; and finally, it extends standard evolutionary dynamics to include phenomena such as the emergence of cooperation and the coexistence of qualitatively different phases of evolution, which we speculate can be associated with Darwinism and punctuated equilibria. * alessandro.bravetti@cimat.mx †

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“…Starting from Gibbs’ fundamental relation, contact geometry has been recognized since the 1970s as an appropriate framework for the geometric formulation of thermodynamics; see in particular [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ]. More recently, the interest in contact-geometric descriptions has been growing, from different points of view and with different motivations; see, e.g., [ 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ].…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, we aim at expanding this symplectization point of view towards thermodynamics, amplifying our initial work [ 24 , 25 ]. In particular, we show how the symplectization point of view not only unifies the energy and entropy representation, but is also very helpful in describing the dynamics of thermodynamic processes, inspired by the notion of the contact control system developed in [ 11 , 12 , 13 , 17 , 18 , 19 ]; see also [ 16 ]. Furthermore, it yields a direct and global definition of a metric on the submanifold describing the state properties, encompassing the locally-defined metrics of Weinhold [ 26 ] and Ruppeiner [ 27 ], and providing a new angle to the equivalence results obtained in [ 3 , 5 , 7 , 10 ].…”

Section: Introductionmentioning

confidence: 99%

Geometry of Thermodynamic Processes

Schaft

Maschke

2018

Entropy

Self Cite

101

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Since the 1970s contact geometry has been recognized as an appropriate framework for the geometric formulation of the state properties of thermodynamic systems, without, however, addressing the formulation of non-equilibrium thermodynamic processes. In [3] it was shown how the symplectization of contact manifolds provides a new vantage point; enabling, among others, to switch between the energy and entropy representations of a thermodynamic system. In the present paper this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, as already largely present in the literature [2,18], appears to be elegant and effective. This culminates in the definition of port-thermodynamic systems, and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control. will be called the vector of co-extensive variables.Following [3] the thermodynamic phase space P(T * Q e ) is defined as the projectivization of T * Q e , i.e., as the fiber bundle over Q e with fiber at any point q e ∈ Q e given by the projective space 1 P(T * q e Q e ). The corresponding projection will be denoted by π : T * Q e → P(T * Q e ). 1Recall that elements of P(T * q e Q e ) are identified with rays in T * q e Q e , i.e., non-zero multiples of a non-zero cotangent vector.

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Partial Stabilization of Input-Output Contact Systems on a Legendre Submanifold

Cited by 48 publications

References 22 publications

Contact Hamiltonian Dynamics: The Concept and Its Use

Contact Hamiltonian Dynamics: The Concept and Its Use

Thermodynamics and evolutionary biology through optimal control

Geometry of Thermodynamic Processes

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