Condensed matter physics, hybrid energy and entropy principles, and the hybrid first and second laws of thermodynamics

Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

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Condensed matter physics, hybrid energy and entropy principles, and the hybrid first and second laws of thermodynamicsR Wassim M. Haddad School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA

a r t i c l e

i n f o

Article history: Received 24 May 2019 Revised 3 October 2019 Accepted 1 November 2019 Available online 6 November 2019 Keywords: Semistability Energy equipartition Hybrid entropy Impulsive energy flow models Hybrid thermodynamics Condensed matter physics Critical phenomena Discontinuous phase transitions

a b s t r a c t In this paper, we develop an energy-based, large-scale hybrid dynamical system model to present a generic framework for hybrid thermodynamics involving hybrid energy and entropy conservation and nonconservation principles. Specifically, using a hybrid compartmental dynamical system energy flow model we develop a hybrid state-space dynamical system formalism for addressing critical phenomena and discontinuous phase transitions in thermodynamics and provide hybrid extensions to the first and second laws of thermodynamics. In addition, using Lyapunov stability theory for impulsive differential equations, we show that our hybrid large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies. Moreover, we show that the steady-state distribution of the hybrid large-scale system energies is uniformly distributed over all of the subsystems, leading to system energy equipartitioning corresponding to a maximum entropy equilibrium state. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In [1], we developed a postmodern framework for thermodynamics that involves open interconnected dynamical systems that exchange matter and energy with their environment in accordance with the first law (conservation of energy) and the second law (nonconservation of entropy) of thermodynamics. In particular, using a large-scale dynamical systems perspective we provide a system-theoretic foundation for equilibrium and nonequilibrium thermodynamics using a state space formulation involving nonlinear compartmental dynamical system models characterized by energy conservation laws that are consistent with basic thermodynamic principles. An implicit assumption of the large-scale dynamical system model developed in [1] is that the thermodynamic state variables define a continuously differentiable flow on the nonnegative orthant of the state space. For systems that possess phase transitions and critical states, this assumption is clearly limiting. The study of phase transitions forms a critical core of structural physics and thermodynamics, and describes transitions between solid, liquid, and gaseous states of matter, and, in rare cases, plasma. More specifically, a phase transition is a phenomenon wherein an abrupt change between phases (i.e., a change between one state of matter into another) occurs as the system parameters (e.g., temperature, pressure, chemical composition, electric field, magnetic field) are varied.

R

This work was supported in part by the Air Force Office of Scientific Research under Grant FA9550-16-0100. E-mail address: [email protected]

https://doi.org/10.1016/j.cnsns.2019.105096 1007-5704/© 2019 Elsevier B.V. All rights reserved.

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W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

Phase transitions are not limited to thermodynamic systems, where temperature or pressure are the phase transition driving parameters. They include, for example, quantum1 (e.g., two-dimensional electron gases), dynamic (e.g., bifurcation), and topological (e.g., structural) phase transitions. In fact, phase transitions are ubiquitous in nature, with the different phases of water (i.e., vapor, liquid, and ice) perhaps being the most familiar, and with transitions involving change from one phase to the other. Other phase transitions include eutectic transformations, peritectic transformations, spinodal decompositions, mesophase transitions, ferromagnetic-paramagnetic phase transitions of magnetic materials, superconductivity, molecular structure (e.g., polymorph, allotrope, polyamorph) transitions, and quantum or Bose-Einstein condensation (e.g., superfluid transition in liquid helium). In a large-class of systems undergoing a phase transition, the system can undergo a symmetry change. For example, in the early universe (i.e., ∼ 10−35 seconds after the cosmic burst and through inflationary cosmology and the initial cosmic horizon), symmetry-breaking transitions in the laws of physics due to ravaging pressures and temperatures prevented the permanent formation of elementary particles (e.g., bosons, quarks, leptons, antiquarks, and antileptons). Elementary and composite particles were unable to form stable constituents until the universe cooled beyond the supergravity phase, that is, gravity as predicted by supersymmetric quantum field theory. These symmetry-breaking phase transitions are critical in understanding the asymmetry between matter and antimatter in the universe (i.e., the cosmic baryon asymmetry). If the phase transition driving parameter is temperature, then the higher temperature phase almost always corresponds to a more disordered state, that is, it has a higher symmetry than the low temperature phase, and hence, phase transitions involve a change in system entropy. For example, a gas has a higher symmetry than a liquid, which in turn has a higher symmetry than a solid [2, pp. 505–508]. And since the maximum entropy of any dynamical system corresponds to state indistinguishability (i.e., total loss of information storage), highest symmetry, and highest simplicity (i.e., decomplexification), entropy increases during transitions from a solid state to a liquid state to a gas state with increasing temperature, and hence, order corresponds to asymmetry, not symmetry2 [3]. This change in entropy, as well as other thermodynamic quantities (e.g., energy, enthalpy, volume), can be continuous or discontinuous. A continuous phase transition is manifest by continuous thermodynamic states with discontinuous first-order derivatives of these states, whereas a discontinuous phase transition is manifest by discontinuous thermodynamic states. The thermodynamic quantities involving a discontinuous jump in systems undergoing a phase transition are known as the system order parameters and the field of condensed matter physics is concerned with identifying and classifying these different phases of matter. For example, solid-liquid and liquid-gas transitions at temperatures below the critical state temperature (i.e., temperatures at which different states of matter coexist) correspond to discontinuous phase transitions, whereas a liquid-gas transition at a temperature above the critical state temperature corresponds to a continuous phase transition. For an excellent exposition of equilibrium and nonequilibrium critical phenomena and phase transitions see [4,5]. In classical thermodynamics, these phase transition states are known as critical states and the theory of critical phenomena addresses this anomalous behavior. In nature, however, thermodynamic critical phenomena occurring at thermal equilibria is the exception and not the rule. In the majority of cases, the system initial state is far from a thermal equilibrium, and hence, specific dynamical properties cannot be described using classical thermodynamics. In such cases, dynamical systems theory can be used to address thermodynamic critical phenomena with continuous and discontinuous phase transitions. As noted above, continuous phase transitions involve a continuous transition across the transition parameter (e.g., a fixed temperature) resulting in continuous changes in the thermodynamic state quantities (e.g., internal energy, entropy, enthalpy, volume) with discontinuous first- and higher-order derivatives of these quantities. These phase transitions are not accompanied by an instantaneous heat release but involve a pronounced anomaly in their specific heat resulting in a very sharp thermal expansion. These transitions are known as second- or higher-order transitions in the literature. In this case, the vector field defining the dynamical system is a discontinuous function of the state and can be characterized by differential inclusions involving Filippov or Krasovskii set-valued maps specifying a set of directions for the state derivative and admitting Filippov or Krasovskii solutions with absolutely continuous curves [6]. Alternatively, phase transitions can also involve a jump discontinuity in the fundamental thermodynamic state quantities, wherein the system is typically in a mixed-phase regime (e.g., boiling of water) and the discontinuous transition is accompanied by an instantaneous heat release (i.e., latent heat). These phase transitions are known as first-order transitions in the literature and can be characterized by dynamical systems with an interacting mixture of continuous and discrete dynamics exhibiting discontinuous flows on appropriate manifolds, giving rise to hybrid dynamics [7,8]. The mathematical descriptions of many hybrid dynamical systems can be modeled by impulsive differential equations [9– 17]. Impulsive dynamical systems can be viewed as a subclass of hybrid systems and consist of three elements: a continuoustime differential equation, which governs the motion of the dynamical system between impulsive or resetting events; a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs; and a criterion for determining when the states of the system are to be reset.

1 Unlike classical phase transitions, quantum phase transitions involve transition driving parameters (e.g., magnetic field, pressure) at an absolute zero temperature. These phase transitions switch between different quantum phases (i.e., phases of matter at absolute zero temperature) and describe an abrupt change in the system ground state involving quantum fluctuations (as opposed to thermal fluctuations) arising from the Heisenberg uncertainty principle. 2 In other words, every isolated dynamical system spontaneously evolves toward a state of maximum entropy, and hence, maximal symmetry, wherein higher entropy (i.e., lower state information content) is connected to higher symmetry.

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Hybrid and impulsive dynamical systems can exhibit a very rich dynamical behavior. In particular, the trajectories of hybrid and impulsive dynamical systems can exhibit multiple complex phenomena, such as Zeno solutions, noncontinuability of solutions or deadlock, beating or livelock, and confluence or merging of solutions [9,14]. A Zeno solution involves a system trajectory with infinitely many resettings in finite time. Deadlock corresponds to a dynamical system state from which no continuation, continuous or discrete, is possible. A hybrid dynamical system experiences beating when the system trajectory encounters the same resetting surface a finite or infinite number of times in zero time. Finally, confluence involves system solutions that coincide after a certain point in time. These phenomena, along with the breakdown of many of the fundamental properties of classical dynamical systems theory, such as continuity of solutions and continuous dependence of solutions on system initial conditions, make the analysis of hybrid and impulsive dynamical systems extremely challenging [8,9,14]. In this paper, we use impulsive dynamical systems to develop a hybrid thermodynamic model as a special case of nonlinear hybrid compartmental and nonnegative dynamical systems to provide a generic framework for thermodynamic critical phenomena with discontinuous phase transitions. 2. Notation, definitions, and mathematical preliminaries In this section, we establish notation and definitions, and review some basic results on stability of impulsive dynamical systems [14]. Specifically, R denotes the set of real numbers, R+ denotes the set of nonnegative real numbers, Rn denotes the n set of n × 1 real column vectors, R+ and Rn+ denote, respectively, the nonnegative and positive orthants of Rn , Z+ denotes T the set of positive integers, ( · ) denotes transpose, and e denotes the ones vector of order n, that is, e = [1, . . . , 1]T . In ◦

addition, we denote the boundary, the interior, and the closure of the set S by ∂ S, S , and S , respectively. We write  ·  for the Euclidean vector norm, Bε (y ), y ∈ Rn , ε > 0, for the open ball centered at y with radius ε , and x(t ) → M as t → ∞ to denote that x(t) approaches the set M (that is, for every ε > 0 there exists T > 0 such that dist(x(t ), M ) < ε for all t > T, where dist( p, M )  infx∈M  p − x). Furthermore, we write V (x) for the Fréchet derivative of V at x. The mathematical foundation for developing hybrid thermodynamic models involves compartmental modeling and nonnegative dynamical systems theory [18,19], which involves dynamical system models characterized by energy conservation laws with nonnegative state variables. In addition, as shown in [1,20], thermodynamic systems give rise to dynamical systems that possess a continuum of equilibria. Since, as we see later in the paper, our thermodynamic state equations govern energy flow between subsystems, it follows from physical considerations that nonnegative system energy initial conditions give rise to dynamical system trajectories that remain in the nonnegative orthant of the state space. Hence, our hybrid thermodynamic model evolves on the nonnegative orthant of the state space and can have the boundary of the orthant as its set of equilibria. In light of the above, we provide sufficient conditions for stability of state-dependent impulsive nonnegative dynamical systems, that is, state-dependent impulsive dynamical systems [14] whose solutions remain in the nonnegative orthant for nonnegative initial conditions. Specifically, we consider nonlinear state-dependent impulsive dynamical systems of the form

x˙ (t ) = fc (x(t )),

x ( 0 ) = x0 ,

x(t ) = fd (x(t )),

x(t ) ∈ Z,

(1)

x(t ) ∈ Z,

(2) n R+

where, for all t ≥ 0, x(t ) ∈ D ⊆ D is a relatively open subset of that contains with 0 ∈ D, x(t )  − x(t ), where x(t + )  x(t ) + fd (x(t )) = limε→0+ x(t + ε ), fc : D → Rn is Lipschitz continuous and satisfies fc (xe ) = 0, fd : D → Rn is continuous, and Z ⊂ D is the resetting set. Note that xe ∈ D is an equilibrium point of (1) and (2) if and only if fc (xe ) = 0 and fd (xe ) = 0. A function x : Ix0 → D is a solution to the impulsive dynamical system (1) and (2) on the interval Ix0 ⊆ R with initial condition x(0 ) = x0 , where Ix0 denotes the maximal interval of existence of a solution (1) and (2), if x( · ) is left-continuous and x(t) satisfies (1) and (2) for all t ∈ Ix0 . For convenience, we use the notation s(t, x0 ) to denote the solution x(t) of (1) and (2) at time t ≥ 0 with initial condition x(0 ) = x0 . We refer to the differential Eq. (1) as the continuous-time dynamics, and we n refer to the difference Eq. (2) as the resetting law. Note that since the resetting set Z is a subset of the state space R+ and is independent of time, state-dependent impulsive dynamical systems are time invariant. For a particular trajectory x(t), we let tk τ k (x0 ) denote the kth instant of time at which x(t) intersects Z, and we call the times τ k (x0 ) the resetting times. Thus, the trajectory of the system (1) and (2) from the initial condition x(0 ) = x0 is given by ψ (t, x0 ) for 0 < t ≤ τ 1 (x0 ), where ψ ( · , · ) denotes the solution to the continuous-time dynamics (1). If the trajectory reaches a state x1  x(τ1 (x0 )) satisfying x1 ∈ Z, then the state is instantaneously transferred to x+  1 x1 + fd (x1 ) according to the resetting law (2). The trajectory x(t), τ 1 (x0 ) < t ≤ τ 2 (x0 ), is then given by ψ (t − τ1 (x0 ), x+ ), and 1 so on. Our convention here is that the solution x(t) of (1) and (2) is left-continuous, that is, it is continuous everywhere except at the resetting times τ k (x0 ), and Rn ,

Rn

x(t + )

xk  x(τk (x0 )) = lim+ x(τk (x0 ) − ε ),

(3)

x+  x(τk (x0 )) + fd (x(τk (x0 ))), k

(4)

ε →0

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for k = 1, 2, . . .. To ensure well-posedness of solutions to (1) and (2), we make the following additional assumptions [14]. A1. If x ∈ Z \Z, then there exists ε > 0 such that, for all 0 < δ < ε , ψ (δ, x ) ∈ Z. A2. If x ∈ Z, then x + fd (x ) ∈ Z. Assumption A1 ensures that if a trajectory reaches the closure of Z at a point that does not belong to Z, then the trajectory must be directed away from Z; that is, a trajectory cannot enter Z through a point that belongs to the closure of Z but not to Z. Furthermore, A2 ensures that when a trajectory intersects the resetting set Z, it instantaneously exits Z. Finally, we note that if x0 ∈ Z, then the system initially resets to x+ = x0 + f d ( x0 ) ∈ / Z, which serves as the initial condition 0 for the continuous dynamics (1). It follows from A1 and A2 that ∂ Z ∩ Z is closed, and hence, the resetting times τ k (x0 ) n are well defined and distinct. Furthermore, it follows from A2 that if x∗ ∈ R+ satisfies fd (x∗ ) = 0, then x∗ ∈ Z. To see this, suppose ad absurdum that x∗ ∈ Z. Then x∗ + fd (x∗ ) = x∗ ∈ Z, contradicting A2. Thus, if x = xe is an equilibrium point of (1) and (2), then xe ∈ Z, and hence, xe ∈ D is an equilibrium point of (1) and (2) if and only if fc (xe ) = 0. Now, since for every x ∈ Z, x + fd (x ) ∈ Z, it follows that τ2 (x ) = τ1 (x ) + τ1 (x + fd (x )) > 0. It follows from Assumptions A1 and A2 that for a particular initial condition, the resetting times tk = τk (x0 ) are distinct and well defined [14]. Hence, since the solution to (1) exists and is unique, it follows that the solution to the impulse dynamical system (1) and (2) also is unique over a forward time interval. Here we assume that if the solution to (1) and (2) is Zeno, then it is convergent and the continuous and discrete parts of the state converge to a unique value at the Zeno time. Note that Assumption A2 precludes the possibility of beating. For further insights on Assumptions A1 and A2 the interested reader is referred to [14]. The following definitions introduce the notions of essentially nonnegative and nonnegative vector fields. n

n

Definition 2.1. Let f = [ f1 , . . . , fn ]T : D ⊆ R+ → Rn . Then f is essentially nonnegative if fi (x) ≥ 0 for all i = 1, . . . , n and x ∈ R+ such that xi = 0, where xi denotes the ith component of x. n

n

Definition 2.2. Let f = [ f1 , . . . , fn ]T : D ⊆ R+ → Rn . Then f is nonnegative if f(x) ≥ ≥ 0 for all x ∈ R+ , where “ ≥ ≥ ” denotes a component-by-component inequality. n

Next, we present a result that shows that R+ is an invariant set with respect to (1) and (2) if fc : D → Rn is essentially n

nonnegative and fd : D → Rn is such that x + fd (x ) is nonnegative for all x ∈ R+ . n

Proposition 2.1. Suppose R+ ⊂ D. If fc : D → Rn is essentially nonnegative and fd : Z → Rn is such that x + fd (x ) is nonnegative, then

n R+

is an invariant set with respect to (1) and (2).

Proof. Consider the continuous-time dynamical system given by

x˙ c (t ) = fc (xc (t )),

xc (0 ) = xc0 ,

t ≥ 0.

(5)

Now, it follows from Proposition 2.1 of [20] that since fc : D → n

n

Rn

is essentially nonnegative,

n R+

is an invariant set with

respect to (5), that is, if xc0 ∈ R+ , then xc (t ) ∈ R+ , t ≥ 0. Now, since, with xc0 = x0 , x(t ) = xc (t ), 0 ≤ t ≤ τ 1 (x0 ), it follows that n

x(t ) ∈ R+ , 0 ≤ t ≤ τ 1 (x0 ). Next, since fd : Z → Rn is such that x + fd (x ) is nonnegative, it follows that x+ = x(τ1 (x0 )) + fd (x(τ1 (x0 ))) ∈ 1 n

n

R+ . Now, since s(t, x0 ) = s(t − τ1 (x0 ), x+ ), τ 1 (x0 ) < t ≤ τ 2 (x0 ), with xc0 = x+ , it follows that x(t ) = xc (t − τ1 (x0 )) ∈ R+ , 1 1

τ 1 (x0 ) < t ≤ τ 2 (x0 ), and hence, x+2 = x(τ2 (x0 )) + fd (x(τ2 (x0 ))) ∈ n

lows that R+ is an invariant set with respect to (1) and (2).

n R+ .

Repeating this procedure for τ i (x0 ), i = 3, 4, . . . , it fol-



It is important to note that, unlike continuous-time and discrete-time nonnegative dynamical systems [18], Proposition n 2.1 provides only sufficient conditions ensuring that R+ is an invariant set with respect to (1) and (2). To see this, let n

n

Z = ∂ R+ and assume x + fd (x ), x ∈ Z, is nonnegative. Then, R+ remains invariant with respect to (1) and (2) irrespective of whether fc ( · ) is essentially nonnegative or not. Next, we present several key results on the stability of nonlinear hybrid nonnegative dynamical systems. Since these systems evolve on closed positively invariant subsets of Rn , the stability of the equilibrium solution of a nonnegative impuln sive dynamical system is defined with respect to perturbed initial conditions that belong to R+ . For addressing the relative stability of impulsive nonnegative dynamical systems the usual relative stability definitions are valid. For a detailed set of n definitions on relative stability with respect to the nonnegative orthant R+ , see [18]. n

Theorem 2.1. Let D be an open subset relative to R+ that contains xe . If there exists a continuously differentiable function V :

n R+

→ [0, ∞ ) satisfying V (xe ) = 0, V(x) > 0, x = xe , and

V  ( x ) f c ( x ) ≤ 0,

x∈ / Z,

V (x + fd (x )) ≤ V (x ),

x ∈ Z,

(6) (7)

then the equilibrium solution x(t) ≡ xe of the hybrid nonnegative dynamical system (1) and (2) is Lyapunov stable with respect n to R+ . Furthermore, if the inequality (6) is strict for all x = xe , then the equilibrium solution x(t) ≡ xe of the hybrid nonnegative

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5

n

dynamical system (1) and (2) is asymptotically stable with respect to R+ . Finally, if V(x) → ∞, as x → ∞, then asymptotic stability with respect to

n R+

is global. n

n

Proof. The proof is identical to the proof of Theorem 2.1 of [14] with D ⊆ R+ and Z ⊂ R+ .



Remark 2.1. For asymptotic stability, Theorem 2.1 requires that the Lyapunov function V( · ) be monotonically decreasing over the continuous-time dynamics and monotonically nonincreasing over the discrete-time dynamics. Less conservative asymptotic stability results have appeared in the literature wherein V( · ) is required to be monotonically nonincreasing everywhere except at the resetting times where the function V( · ) is only allowed to decrease. In addition, stronger asymptotic stability theorems are also provided in the literature that only require that the right limits of V( · ) at the resetting times be nonincreasing and at all other times between resettings V( · ) be bounded by the combination of a prespecified bounded function and the right limit of V( · ) at the resetting time. For details, see [8,21]. Next, we present generalized Krasovskii–LaSalle invariant set stability theorems for nonlinear hybrid dynamical systems. The following key assumptions and propositions are needed for the statement of these results. For the first assumption, define Tx0  [0, ∞ )\{τ1 (x0 ), τ2 (x0 ), . . .}. Assumption 2.1 [14]. Consider the impulsive nonnegative dynamical system G given by (1) and (2), and let s(t, x0 ), t ≥ 0, denote the solution to (1) and (2) with initial condition x0 . Then, for every x0 ∈ D, there exists a dense subset Tx0 ⊆ [0, ∞ ) such that [0, ∞ ) \ Tx0 is (finitely or infinitely) countable and, for every ε > 0 and t ∈ Tx0 , there exists δ (ε , x0 , t) > 0 such that if x0 − y < δ (ε , x0 , t ), y ∈ D, then s(t, x0 ) − s(t, y ) < ε . Assumption 2.1 is a quasi-continuous dependence property and is a generalization of the standard continuous dependence property for dynamical systems with continuous flows to dynamical systems with left-continuous flows and ensures continuous dependence over a dense subset of [0, ∞). Assumption 2.1 is key in guaranteeing invariance of positive limit sets for state-dependent impulsive dynamical systems. However, as discussed in [14], in order to guarantee invariance of positive limit sets for state-dependent impulsive dynamical systems, the quasi-continuous dependence property (Assumption 2.1) need only be satisfied for trajectories starting on the positive limit set. Since it is generally difficult to verify that the quasi-continuous dependence property holds for trajectories on the positive limit set, in Assumption 2.1 we assume that the quasi-continuous dependence property holds for every trajectory in D. In practice, however, this assumption can be restrictive. Next, we weaken Assumption 2.1 by assuming point-wise continuous dependence. As shown in [14], this weakened version of Assumption 2.1 is sufficient for guaranteeing invariance of positive limit sets for a special class of state-dependent impulsive dynamical systems. Specifically, we consider impulsive dynamical systems of the form (1) and (2) for which x0 ∈ Z implies that x0 + fd (x0 ) ∈ Z \Z. In this case, it can be shown that the positive limit sets of all trajectories of (1) and (2) lie on Z \Z. Hence, we need only assume quasi-continuous dependence for trajectories starting outside Z. Assumption 2.2 [14]. Consider the impulsive dynamical system (1) and (2), and let s(t, x0 ), t ≥ 0, denote the solution to (1) and (2) with initial condition x0 . Then for every x0 ∈ Z and every ε > 0 and t = tk , there exists δ (ε , x0 , t) > 0 such that if x0 − z < δ (ε, x0 , t ), z ∈ D, then s(t, x0 ) − s(t, z ) < ε. The following result provides sufficient conditions that guarantee that the nonlinear impulsive dynamical system G given by (1) and (2) satisfies Assumption 2.2. For further discussion on Assumptions 2.1 and 2.2, see [14]. Proposition 2.2 [14]. Consider the impulsive dynamical system G given by (1) and (2). Assume that Assumptions A1 and A2 hold, τ 1 ( · ) is continuous at every x ∈/ Z such that 0 < τ 1 (x) < ∞, and if x ∈ Z, then x + fd (x ) ∈ Z \Z. Furthermore, for every x ∈ Z \Z such that 0 < τ 1 (x) < ∞, assume that the following statements hold: i) If a sequence {xi }∞ ∈ D is such that limi→∞ xi = x and limi → ∞ τ 1 (xi ) exists, then either fd (x ) = 0 and limi→∞ τ1 (xi ) = 0, or i=1 limi→∞ τ1 (xi ) = τ1 (x ). ii) If a sequence {xi }∞ ∈ Z \Z is such that limi→∞ xi = x and limi → ∞ τ 1 (xi ) exists, then limi→∞ τ1 (xi ) = τ1 (x ). i=1 Then G satisfies Assumption 2.1. The following result provides sufficient conditions for establishing continuity of τ 1 ( · ) at x0 ∈ / Z and sequential continuity of τ 1 ( · ) at x0 ∈ Z \Z, that is, limi→∞ τ1 (xi ) = τ1 (x0 ) for {xi }∞ ∈ / Z and limi→∞ xi = x0 . For this result, the following i=1 definition is needed. First, however, recall that the Lie derivative of a smooth function X : D → R along the vector field of the continuous-time dynamics fc (x) is given by L fc X (x )  ddt X (ψ (t, x ))|t=0 = ∂ X∂ (xx ) fc (x ), and the zeroth and higher-order Lie derivatives are, respectively, defined by L0f X (x )  X (x ) and Lkf X (x )  L fc (Lkf −1 X (x )), where k ≥ 1. c

c

c

Definition 2.3. Let Q  {x ∈ D : X (x ) = 0}, where X : D → R is an infinitely differentiable function. A point x ∈ Q such that fc (x) = 0 is k-transversal to (1) if there exists k ∈ {1, 2, . . .} such that

Lrfc X (x ) = 0,

r = 0, . . . , 2k − 2,

L2fck−1 X (x ) = 0.

(8)

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W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

Proposition 2.3 [14]. Consider the impulsive dynamical system (1) and (2). Let X : D → R be an infinitely differentiable function such that Z = {x ∈ D : X (x ) = 0}, and assume that every x ∈ Z is k-transversal to (1). Then at every x0 ∈ /Z such that 0 < τ 1 (x0 ) < ∞, τ 1 ( · ) is continuous. Furthermore, if x0 ∈ Z \Z is such that τ 1 (x0 ) ∈ (0, ∞) and {xi }∞ ∈ Z \Z or i=1 limi → ∞ τ 1 (xi ) > 0, where {xi }∞ ∈ / Z is such that limi→∞ xi = x0 and limi → ∞ τ 1 (xi ) exists, then limi→∞ τ1 (xi ) = τ1 (x0 ). i=1 Note that if x0 ∈ Z is such that limi → ∞ τ 1 (xi ) = τ 1 (x0 ) for some sequence {xi }∞ ∈ Z, then it follows from i=1 Proposition 2.3 that limi→∞ τ1 (xi ) = 0. The notion of k-transversality introduced in Definition 2.3 differs from the wellknown notion of transversality [22,23] involving an orthogonality condition between a vector field and a differentiable submanifold. In the case where k = 1, Definition 2.3 coincides with the standard notion of transversality and guarantees that the solution of the system (1) and (2) is not tangent to the closure of the resetting set Z at the intersection with Z [14]. In general, however, k-transversality guarantees that the sign of X (x(t )) changes as the dynamical system trajectory transverses the closure of the resetting set Z at the intersection with Z . n Note that for an impulsive dynamical system a point p ∈ D ⊆ R+ is a positive limit point of the trajectory s(t, x0 ), t ≥ 0, ∞ if and only if there exists a monotonic sequence {tn }n=0 ⊂ Tx0 , with tn → ∞ as n → ∞, such that s(tn , x0 ) → p as n → ∞. To see this, let p ∈ ω(x0 ), where ω(x0 ) denotes the positive limit set of G, and recall that Tx0 is a dense subset of the semiinfinite interval [0, ∞). In this case, it follows that there exists an unbounded sequence {tn }∞ , with tn → ∞ as n → ∞, n=0 such that limn→∞ s(tn , x0 ) = p. Hence, for every ε > 0, there exists n > 0 such that s(tn , x0 ) − p < ε /2. Furthermore, since s( · , x0 ) is left-continuous and Tx0 is a dense subset of [0, ∞), there exists tˆn ∈ Tx0 , tˆn ≤ tn , such that s(tˆn , x0 ) − s(tn , x0 ) < ε/2, and hence, s(tˆn , x0 ) − p ≤ s(tn , x0 ) − p + s(tˆn , x0 ) − s(tn , x0 ) < ε. Using this procedure, with ε = 1, 1/2, 1/3, . . . , we can construct an unbounded sequence {tˆk }∞ ⊂ Tx0 such that limk→∞ s(tˆk , x0 ) = p. Hence, p ∈ ω(x0 ) if and only if there k=1 exists a monotonic sequence {tn }∞ ⊂ T , with t → ∞ as n → ∞, such that s(tn , x0 ) → p as n → ∞. x n 0 n=0 Next, we state two versions of the Krasovskii–LaSalle invariance principle for state-dependent impulsive dynamical systems. These results characterize impulsive dynamical system limit sets in terms of continuously differentiable functions. In particular, the results show that the system trajectories converge to an invariant set contained in a union of level surfaces characterized by the continuous-time dynamics and the resetting system dynamics. For the first result, we assume that fc ( · ), fd ( · ), and Z are such that the dynamical system G given by (1) and (2) satisfies Assumptions A1, A2, and 2.1. n

Theorem 2.2. Consider the hybrid nonnegative dynamical system G given by (1) and (2), assume Dc ⊂ D ⊆ R+ is a compact positively invariant set with respect to (1) and (2), and assume that there exists a continuously differentiable function V : Dc → R such that

V  ( x ) f c ( x ) ≤ 0,

x ∈ Dc ,

V (x + fd (x )) ≤ V (x ),

x∈ / Z,

x ∈ Dc ,

(9)

x ∈ Z.

(10)

Let R  {x ∈ Dc : x ∈ / Z, V  (x ) fc (x ) = 0} ∪

{x ∈ Dc : x ∈ Z, V (x + fd (x )) = V (x )} and let M denote the largest invariant set contained in R. If x0 ∈ Dc , then x(t ) → M as t → ∞. n

Proof. The proof is a direct consequence of Theorem 2.3 of [14] with Dc ⊂ D ⊆ R+ .



For the next result, we assume that fc ( · ), fd ( · ), and Z are such that the dynamical system G given by (1) and (2) satisfies Assumptions A1, A2, and 2.2, and Z ∩ {x : fd (x ) = x} is empty. n

Theorem 2.3. Consider the hybrid nonnegative dynamical system G given by (1) and (2), assume Dc ⊂ D ⊆ R+ is a compact positively invariant set with respect to (1) and (2), assume that if x0 ∈ Z, then x0 + fd (x0 ) ∈ Z \Z, and assume that there exists a continuously differentiable function V : Dc → R such that

V  ( x ) f c ( x ) ≤ 0,

x ∈ Dc ,

V (x + fd (x )) ≤ V (x ),

x∈ / Z,

x ∈ Dc ,

(11)

x ∈ Z.

(12)

Let R  {x ∈ Dc : x ∈ / Z, V  (x ) fc (x ) = 0} ∪ {x ∈ Dc : x ∈ Z, V (x + fd (x )) = V (x )} and let M denote the largest invariant set contained in R. If x0 ∈ Dc , then x(t ) → M as t → ∞. n

Proof. The proof is a direct consequence of Theorem 8.1 of [14] with Dc ⊂ D ⊆ R+ .



3. Hybrid thermodynamic models In this section, we develop a hybrid thermodynamic system model. To formulate our state space hybrid thermodynamic model, let Ei (t), i = 1, . . . , q, denote the energy (and hence a nonnegative quantity3 ) of the ith subsystem of the hybrid compartmental system shown in Fig. 3.1, let σ cii (E) ≥ 0, E ∈ / Z, where E  [E1 , . . . , Eq ]T , denote the rate of flow of energy 3 Here we assume that subsystem energies are lower bounded so that, without loss of generality, we can shift Ei ( · ) such that, with a minor abuse of notation, Ei (t) ≥ 0, t ≥ 0, i = 1, . . . , q. This assumption corresponds to the existence of a minimal energy state (i.e., the ground or vacuum state).

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

7

Fig. 3.1. Nonlinear hybrid energy flow model.

loss of the ith continuous-time subsystem, let Sci (t) ≥ 0, t ≥ 0, i = 1, . . . , q, denote the rate of energy inflow supplied to the ith continuous-time subsystem, and let φ cij (E(t)), t ≥ 0, i = j, i, j = 1, . . . , q, denote the net energy flow (or power) from the jth continuous-time subsystem to the ith continuous-time subsystem given by φci j (E (t )) = σci j (E (t )) − σc ji (E (t )), where the rates of energy flows are such that σci j (E ) ≥ 0, E ∈ / Z, i = j, i, j = 1, . . . , q. Similarly, for the resetting dynamics, let σ dii (E) ≥ 0, E ∈ Z, denote the energy loss of the ith discrete-time subsystem, let Sdi (tk ) ≥ 0, i = 1, . . . , q, denote the energy inflow supplied to the ith discrete-time subsystem, and let φ dij (tk ), i = j, i, j = 1, . . . , q, denote the net energy exchange from the jth discrete-time subsystem to the ith discrete-time subsystem given by φdi j (E (tk )) = σdi j (E (tk )) − σd ji (E (tk )), where tk = τk (E0 ) and the energy flows are such that σ dij (E) ≥ 0, E ∈ Z, i = j, i, j = 1, . . . , q. It is important to note that the exchange of energy between subsystems φ cij (E) and φ dij (E) is assumed to be a nonlinear q

function of all the subsystem energies, that is, φci j = φci j (E ) and φdi j = φdi j (E ), E ∈ R+ , i = j, i, j = 1, . . . , q. This assumption is made for generality and would depend on the complexity of the energy flow process. For example, thermal processes may include evaporative and radiative heat transfer as well as thermal conduction giving rise to complex heat transport mechanisms. Alternatively, for simple diffusion processes it suffices to assume that φci j (E ) = φci j (Ei , E j ) and φdi j (E ) = φdi j (Ei , E j ), wherein the energy exchange between any two subsystems is only dependent (possibly nonlinearly) on the energies of the two subsystems. Furthermore, the resetting set Z will depend on the type of thermodynamic phase transition as well as the phase transition driving parameters (e.g., system temperature and/or pressure). For example, a solid-liquid transition will involve melting and freezing effects; a liquid-gas transition will involve vaporization and condensation effects; a solid-gas transition will involve sublimation and deposition effects; whereas a gas-plasma transition will involve ionization and recombination effects. In all of the aforementioned phase transitions the system entropy can discontinuously increase. Here, we let the resetting set Z be generic capturing any of the aforementioned phase transitions and we assume that Z is such that Assumptions A1, A2, and 2.1 (or 2.2) hold. For a further discussion on the structure of Z, see Section VII. Next, an energy balance for the whole hybrid compartmental thermodynamic system yields

E˙ i (t ) = −σcii (E (t )) +

q 

φci j (E (t )) + Sci (t ), E (t ) ∈ Z, i = 1, . . . , q,

(13)

j=1,i = j q 

Ei (t ) = −σdii (E (t )) +

φdi j (E (t )) + Sdi (t ), E (t ) ∈ Z, i = 1, . . . , q,

(14)

j=1,i = j

or, equivalently, in verctor form

E˙ (t ) = fc (E (t )) + Sc (t ),

E ( 0 ) = E0 ,

E (t ) = fd (E (t )) + Sd (t ), where E (t )  [E1 (t ), . . . , Eq

(t )]T , 

E (t ) ∈ Z,

(15)

E (t ) ∈ Z,

Sc (t )  [Sc1 (t ), . . . , Scq

(16)

(t )]T ,

Sd (t )  [Sd1 (t ), . . . , Sdq

(t )]T ,

and, for i, j = 1, . . . , q,

q

fci (E ) = −σcii (E ) +

[σci j (E ) − σc ji (E )],

(17)

[σdi j (E ) − σd ji (E )].

(18)

j=1,i = j

fdi (E ) = −σdii (E ) +

q  j=1,i = j

Since all energy flows as well as compartment sizes are nonnegative, it follows that for all i = 1, . . . , q, fci (E) ≥ 0 for all E ∈ Z, whenever Ei = 0 and whatever the values of Ej , j = i, and Ei + fdi (E ) ≥ 0 for all E ∈ Z. The above physical constraints

8

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

are implied by σ cij (E) ≥ 0, σ cii (E) ≥ 0, E ∈ Z, σ dij (E) ≥ 0, σ dii (E) ≥ 0, E ∈ Z, Sci ≥ 0, Sdi ≥ 0, for all i, j = 1, . . . , q, and if Ei = 0, then σcii (E ) = 0 and σc ji (E ) = 0 for all i, j = 1, . . . , q, so that E˙ i ≥ 0. In this case, fc (E ), E ∈ / Z, is essentially nonnegative and E + fd (E ) ≥≥ 0, E ∈ Z, and hence, it follows from Proposition 2.1 that the hybrid compartmental model given by (13) and (14) is a hybrid nonnegative dynamical system. q Taking the total energy of the compartmental system V (E ) = eT E = i=1 Ei as a Lyapunov function candidate for the inflow-closed (i.e., Sc (t) ≡ 0 and Sd (tk ) ≡ 0) system (13) and (14), and assuming σi j (0 ) = 0, i, j = 1, . . . , q, it follows that q 

V˙ (E ) =

E˙ i

i=1

=−

q 

σcii (E ) +

i=1

=−

q 

q q  

[σci j (E ) − σc ji (E )]

i=1 j=1,i = j

σcii (E )

i=1

≤ 0,

E ∈ Z,

and

V (E ) =

q 

E i

i=1

=−

q 

σdii (E ) +

i=1

=−

q 

q q  

[σdi j (E ) − σd ji (E )]

i=1 j=1,i = j

σdii (E )

i=1

≤ 0,

E ∈ Z,

which, by Theorem 2.1, shows that the zero solution E(t) ≡ 0 of the nonlinear hybrid compartmental system given by q (13) and (14) is Lyapunov stable with respect to R+ . If (13) and (14) with Sc (t) ≡ 0 and Sd (tk ) ≡ 0 has energy losses (outflows) from all compartments over the continuous-time dynamics, then σ cii (E) > 0, E ∈ Z, E = 0, and hence, by Theorem 2.1, q the zero solution E(t) ≡ 0 to (13) and (14) is asymptotically stable with respect to R+ . It is interesting to note that in the linear case σcii (E ) = σcii Ei , φci j (E ) = σci j E j − σc ji Ei , σdii (E ) = σdii Ei , and φdi j (E ) = σdi j E j − σd ji Ei , where σ cij ≥ 0 and σ dij ≥ 0, i, j = 1, . . . , q, (15) and (16) become

E˙ (t ) = Ac E (t ) + Sc (t ),

E ( 0 ) = E0 ,

E (t ) = (Ad − In )E (t ) + Sd (t ),

E (t ) ∈ Z,

(19)

E (t ) ∈ Z,

(20)

where, for i, j = 1, . . . , q,

 q

Ac(i, j ) = Ad(i, j ) =

−

l=1

σc i j ,  q 1−

σd i j ,

σcli ,

l=1

σdli ,

i = j, i = j,

(21)

i = j, i = j.

(22)

Note that, since at any given instant of time compartmental energy can only be transported, stored, or discharged but not created and the maximum amount of energy that can be transported and/or discharged cannot exceed the energy in a q compartment, it follows that 1 ≥ l=1 σdli . Thus, Ac is an essentially nonnegative matrix (i.e., Ac(i, j )) ≥ 0, i, j = 1, . . . , q, i = j) and Ad is a nonnegative matrix (i.e., Ad(i, j ) ≥ 0, i, j = 1, . . . , q), and hence, the hybrid compartmental model given by (19) and (20) is a hybrid nonnegative dynamical system. The hybrid compartmental thermodynamic system (13) and (14) with no energy inflows, that is, Sci (t) ≡ 0 and Sdi (tk ) ≡ 0, i = 1, . . . , q, is said to be inflow-closed. Alternatively, if (13) and (14) possesses no energy losses (outflows), it is said to be outflow-closed. A hybrid compartmental system is said to be adiabatically isolated if it is inflow-closed and outflow-closed. Note that for an adiabatically isolated system V˙ (E ) = 0, E ∈ Z, and V (E ) = 0, E ∈ Z, which shows that the total energy inside an adiabatically isolated system is conserved. In the case where σcii (E ) = 0, E ∈ / Z, σdii (E ) = 0, E ∈ Z, Sci (t) = 0, and Sdi (tk ) = 0, i = 1, . . . , q, it follows that (13) and (14) can be equivalently written as



E˙ (t ) = [Jcn (E (t )) − Dc (E (t ))]

∂V (E (t )) ∂E

T

+ Sc (t ),

E (t ) ∈ Z,

(23)

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

 E (t ) = [Jdn (E (t )) − Dd (E (t ))]

∂V (E (t )) ∂E

9

T + Sd (t ),

E (t ) ∈ Z,

(24)

where Jcn (E) and Jdn (E) are skew-symmetric matrix functions with

Jcn(i,i ) (E ) = 0,

i = 1, . . . , q,

Jdn(i,i ) (E ) = 0,

i = 1, . . . , q,

σci j (E ) − σc ji (E ), i = j, Jdn(i, j ) (E ) = σdi j (E ) − σd ji (E ), i = j, Jcn(i, j ) (E ) =

Dc (E ) = diag[σc11 (E ), . . . , σcnn (E )] ≥≥ 0,

n

E ∈ R+ ,

and

Dd (E ) = diag[σd11 (E ), . . . , σdnn (E )] ≥≥ 0,

n

E ∈ R+ .

Hence, a hybrid compartmental thermodynamic system is a hybrid port-controlled Hamiltonian system [24] with a Hamiln tonian function H (E ) = V (E ) = eT E representing the total energy in the system, Jcn (E), E ∈ R+ , representing the energy exn

change between subsystems over the continuous-time dynamics, Jdn (E), E ∈ R+ , representing the energy exchange between subsystems at the resetting instants, Dc (E), E ∈ n

n R+ ,

representing the energy dissipation over the continuous-time dynam-

ics, Dd (E), E ∈ R+ , representing the energy dissipation at the resetting instants, Sc (t) representing the supplied power to the system over the continuous-time dynamics, and Sd (tk ) representing the supplied energy to the system at the resetting instants. Finally, we show that our hybrid compartmental thermodynamic system with measured outputs corresponding to energy and energy rate outflows are lossless [14] with respect to the hybrid energy supply rate (rc (uc , yc ), rd (ud , yd )) = (eT uc − eT yc , eT ud − eT yd ), where uc (t )  Sc (t ), yc (t )  Dc (E (t ))e, ud (tk )  Sd (tk ), and yd (tk )  Dd (E (tk ))e. Specifically, consider (23) and (24) with Sc (t ) = uc (t ) and Sd (tk ) = ud (tk ), energy function V (E ) = eT E, and hybrid outputs

 yc = Dc ( E )

∂V ∂E

T = [σc11 (E ), σc22 (E ), . . . , σcnn (E )]T

and



∂V yd = Dd ( E ) ∂E

T = [σd11 (E ), σd22 (E ), . . . , σdnn (E )]T .

Now, it follows that





V˙ (E ) = eT [Jcn (E ) − Dc (E )]

∂V ∂E



T + uc

= eT uc − eT yc + eT Jcn (E )e = eT uc − eT yc , and



E ∈ Z,

(25)



∂V V (E ) = eT [Jdn (E ) − Dd (E )] ∂E



T + ud

= eT ud − eT yd + eT Jdn (E )e = e T u d − eT y d ,

E ∈ Z,

(26)

which shows that the hybrid thermodynamic system (23) and (24) is lossless with respect to the hybrid supply rate (rc , rd ) = (eT Sc − eT yc , eT Sd − eT yd ) [14]. Alternatively, if the hybrid outputs yc and yd correspond to a partial observation of the energy and energy rate outflows, then it can easily be shown that the hybrid compartmental thermodynamic system is dissipative with respect to the hybrid supply rate (rc , rd ) = (eT uc − eT yc , eT ud − eT yd ). The notion of hybrid dissipativity provides an interpretation of a generalized hybrid energy balance for a hybrid dynamical system in terms of the stored or accumulated energy, dissipated energy over the continuous-time dynamics, and dissipated energy at the resetting instants. For a detailed development of hybrid lossless and dissipativity theory, see [14].

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W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

4. Conservation of energy and the hybrid first law of thermodynamics In this section, we develop a hybrid version of the first law of thermodynamics and establish the uniqueness of the q internal energy function U (E ) = V (E ) = eT E, E ∈ R+ , for our hybrid thermodynamic model



T ∂V (E (t )) + Sc (t ), E (t ) ∈ Z, ∂E  T ∂V E (t ) = [Jdn (E (t )) − Dd (E (t ))] (E (t )) + Sd (t ), E (t ) ∈ Z, ∂E  T ∂V yc (t ) = Dc (E (t )) (E (t )) , E (t ) ∈ Z, ∂E  T ∂V yd (t ) = Dd (E (t )) (E (t )) , E (t ) ∈ Z, ∂E E˙ (t ) = [Jcn (E (t )) − Dc (E (t ))]

q

(27)

(28)

(29)

(30) q

where, for all t ≥ 0, E (t ) ∈ D ⊆ R+ , D is a relatively open set with respect to R+ and with 0 ∈ D, E (t )  E (t + ) − E (t ), Sc (t )  uc (t ) ∈ Uc ⊆ Rq , Sd (tk )  ud (tk ) ∈ Ud ⊆ Rq , tk denotes the kth instant of time at which (E(t), uc (t)) intersects Z ⊂ D × Uc for a particular trajectory E(t) and input uc (t), yc (t ) ∈ Yc ⊆ Rq , yd (tk ) ∈ Yd ⊆ Rq , and Z ⊂ D × Uc . Here, we assume that uc ( · ) and ud ( · ) are restricted to the class of admissible inputs consisting of measurable functions such that (uc (t), ud (tk )) ∈ Uc × Ud for all t ≥ 0 and k ∈ Z[0,t )  {k : 0 ≤ tk < t }, where the constraint set Uc × Ud is given with (0, 0) ∈ Uc × Ud . Furthermore, we assume that the set Z  {(E, uc ) : X (E, uc ) = 0}, where X : D × Uc → R. q More precisely, for the hybrid dynamical system G given by (27)–(30) defined on the state space D ⊆ R+ , the spaces U  Uc × Ud and Y  Yc × Yd define an input and output space, respectively, consisting of left-continuous bounded U-valued and Y-valued functions on the semi-infinite interval [0, ∞). The set U  Uc × Ud , where Uc ⊆ Rq and Ud ⊆ Rq , contains the set of input values, that is, for every u = (uc , ud ) ∈ U and t ∈ [0, ∞), u(t) ∈ U, uc (t) ∈ Uc , and ud (tk ) ∈ Ud . The set Y  Yc × Yd , where Yc ⊆ Rlc and Yd ⊆ Rld , contains the set of output values, that is, for every y = (yc , yd ) ∈ Y and t ∈ [0, ∞), y(t) ∈ Y, yc (t) ∈ Yc , and yd (tk ) ∈ Yd . Furthermore, we let Ur denote the set of all bounded continuous and discrete inputs (heat fluxes and energy flows) u = (uc , ud ) to the hybrid large-scale dynamical system G such that, for every T ≥ −t0 , the system energy state can be driven q from E (−T ) = 0 to E (t0 ) = E0 ∈ R+ by u(· ) ∈ Ur . The spaces U and Y are assumed to be closed under the shift operator, that is, if u(· ) ∈ U (respectively, y(· ) ∈ Y), then the function uT (respectively, yT ) defined by uT  u(t + T ) (respectively, yT  y(t + T )) is contained in U (respectively, Y) for all T ≥ 0. It follows from Lemma 3.1 of [1] and Proposition 3.1 of [25] that the hybrid large-scale dynamical system G given by q q (27)–(30) is reachable from and controllable to the origin in R+ . In particular, for all E0 = E (t0 ) ∈ R+ , there exist a finite time ti ≤ t0 , a square integrable input uc (t) defined on [ti , t0 ], and an input ud (tk ) defined on k ∈ Z[ti ,t0 ) , such that the state q

E(t), t ≥ ti , can be driven from E (ti ) = 0 to E (t0 ) = E0 . Furthermore, for all E0 = E (t0 ) ∈ R+ , there exists a finite time tf ≥ t0 , a square integrable input uc (t) defined on [t0 , tf ], and an input ud (tk ) defined on k ∈ Z[ti ,t0 ) , such that the state E(t), t ≥ t0 , can be driven from E (t0 ) = E0 to E (tf ) = 0. n Next, note that with U (E ) = V (E ) = eT E, E ∈ R+ , (25) and (26) are a statement of the first law of thermodynamics as applied to hybrid isochoric transformations (i.e., constant subsystem volume transformations). Specifically, note that (25) and (26) can be written as the single lossless condition



U (E (T )) = U (E (t0 )) +

T

t0

rc (uc (t ), yc (t ))dt +

 k∈Z[t

rd (ud (tk ), yd (tk )),

(31)

0 ,T )

where E(t), t ≥ t0 , is a solution to (27)–(30) with (uc (t), ud (tk )) ∈ Uc × Ud and E (t0 ) = E0 . To see this, next we give necessary and sufficient conditions for losslessness over an interval t ∈ (tk , tk+1 ] involving the consecutive resetting times tk and tk+1 . Theorem 4.1. Consider the hybrid large-scale dynamical system G given by (27)–(30). Then G is lossless with respect to the n hybrid supply rate (rc , rd ) and with energy storage function U : R+ → R+ if and only if, for all k ∈ Z+ ,

U (E (tˆ)) − U (E (t )) =



tˆ

rc (uc (s ), yc (s ))ds,

t

tk < t ≤ tˆ ≤ tk+1 ,

U (E (tk+ )) − U (E (tk )) = rd (ud (tk ), yd (tk )).

(32) (33)

Proof. Let k ∈ Z+ and suppose G is lossless with respect to the hybrid supply rate (rc , rd ) and with energy storage function n U : R+ → R+ . Then (31) holds. Now, since for tk < t ≤ tˆ ≤ tk+1 , Z[t,tˆ) = ∅, (32) is immediate. Next, note that

U (E (tk+ )) − U (E (tk )) ≤



tk+

tk

rc (uc (s ), yc (s ))ds + rd (ud (tk ), yd (tk )),

(34)

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

which, since Z[t

+ k ,tk )

11

= {k}, implies (33).

Conversely, suppose (32) and (33) hold, let tˆ ≥ t ≥ 0, and let Z[t,tˆ) = {i, i + 1, . . . , j}. (Note that if Z[t,tˆ) = ∅, then the converse is a direct consequence of (31).) In this case, it follows from (32) and (33) that

U (E (tˆ)) − U (E (t )) = U (E (tˆ)) − U (E (t + )) + U (E (t +j )) − U (E (t +j−1 )) j + U (E (t + )) − · · · − U (E (ti+ )) j−1 + U (E (ti+ )) − U (E (t )) tˆ = rc (uc (s ), yc (s ))ds + rd (ud (t j ), yd (t j )) t+ j



+

+ =

tj

rc (uc (s ), yc (s ))ds + · · · + rd (ud (ti ), yd (ti ))

t+ j−1 ti

rc (uc (s ), yc (s ))ds

t tˆ

t

rc (uc (s ), yc (s ))ds +



rd (ud (tk ), yd (tk )),

k∈Z[t,tˆ)

which implies that G is lossless with respect to the hybrid supply rate (rc , rd ).



If U(E( · )) is continuously differentiable almost everywhere on [t0 , ∞) except on an unbounded closed discrete set T =

{t1 , t2 , . . .}, where T is the set of times when jumps occur for E(t), then it follows from Theorem 4.1 that an equivalent statement for losslessness of the hybrid large-scale dynamical system G with respect to the hybrid supply rate (rc , rd ) is

U˙ (E (t )) = rc (uc (t ), yc (t )),

tk < t ≤ tk+1 ,

U (E (tk )) = rd (ud (tk ), yd (tk )),

(35)

k ∈ Z+ ,

(36)

where U˙ (· ) denotes the total derivative of U(E(t)) along the state trajectories E(t), t ∈ (tk , tk+1 ], of the hybrid large-scale dynamical system (27)–(30) and U (E (tk ))  U (E (tk+ )) − U (E (tk )), k ∈ Z+ , denotes the difference of the energy function U(E) at the resetting times tk , k ∈ Z+ , of the hybrid large-scale dynamical system (27)–(30). n The next result establishes the uniqueness of the internal energy function U (E ) = eT E, E ∈ R+ , for the hybrid large-scale dynamical system G given by (27)–(30). For this result define the hybrid available energy Ua (E0 ) of the hybrid dynamical system G by

Ua (E0 )  −



inf

(uc (· ),ud (· )), T ≥t0

t0

=

T

sup

(uc (· ),ud (· )), T ≥t0

−

rc (uc (t ), yc (t ))dt +

k∈Z[t T

t0





rc (uc (t ), yc (t ))dt −

rd (ud (tk ), yd (tk ))

0 ,T )

 k∈Z[t

 rd (ud (tk ), yd (tk )) ,

(37)

0 ,T )

where E(t), t ≥ t0 , is the solution to (27)–(30) with admissible inputs (uc (· ), ud (· )) ∈ Uc × Ud and E (t0 ) = E0 . Note that Ua (E0 ) ≥ 0 for all E0 ∈ D since Ua (E0 ) is the supremum over a set of numbers containing the zero element (T = t0 ). It follows from (37) that the hybrid available energy of a hybrid large-scale dynamical system G is the maximum amount of stored energy that can be extracted from G at any time T. Furthermore, note that Ua (E(t)) is left-continuous on [t0 , ∞) and is continuous everywhere on [t0 , ∞) except on an unbounded closed discrete set T = {t1 , t2 , . . . , }, where T is the set of times when the jumps occur for E(t), t ≥ t0 . In addition, define the hybrid required supply Ur (E0 ) of the hybrid large-scale dynamical system G by

Ur (E0 ) 



inf

(uc (· ),ud (· )), T ≤t0

T

t0

rc (uc (t ), yc (t ))dt +





rd (ud (tk ), yd (tk )) ,

(38)

k∈Z[T,t ) 0

where E(t), t ≥ T, is the solution of (27)–(30) with E (T ) = 0 and E (t0 ) = E0 . It follows from (38) that the hybrid required supply of the dynamical system G is the minimum amount of energy that can be delivered to the hybrid dynamical system in order to transfer it from an initial state E (T ) = 0 to a given state E (t0 ) = E0 . Theorem 4.2. Consider the large-scale hybrid dynamical system G given by (27)–(30). Then G is lossless with respect to the hybrid energy supply rate (rc (uc , yc ), rd (ud , yd )) = (eT uc − eT yc , eT ud − eT yd ), where uc (t )  Sc (t ), yc (t )  Dc (E (t ))e, ud (tk )  Sd (tk ), and yd (tk )  Dd (E (tk ))e, and with the unique energy storage function corresponding to the total energy of the system G given by

U ( E0 ) = eT E0

12

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

=−

t0

=

T+

t0

Z[t

rd (ud (tk ), yd (tk ))

0 ,T+ )



rc (uc (t ), yc (t ))dt +

−T−



rc (uc (t ), yc (t ))dt −

rd (ud (tk ), yd (tk )),

(39)

Z[−T− ,t ) 0

where E(t), t ≥ t0 , is the solution to (27)–(30) with admissible hybrid inputs u(· ) = (uc (· ), ud (· )) ∈ U, E (−T− ) = 0, E (T+ ) = 0, q and E (t0 ) = E0 ∈ R+ . Furthermore, q

0 ≤ Ua (E0 ) = U (E0 ) = Ur (E0 ) < ∞,

E 0 ∈ R+ .

(40)

Proof. Note that it follows from (31) that G is lossless with respect to the hybrid supply rate (rc , rd ) = (eT uc − eT yc , eT ud − q eT yd ) and with the energy storage function U (E ) = eT E, E ∈ R+ . Since, by Lemma 3.1 of [1] and Proposition 3.1 of [25], G q

q

is reachable from and controllable to the origin in R+ , it follows from (31), with E (t0 ) = E0 ∈ R+ and E (T+ ) = 0 for some T+ ≥ t0 and u(· ) ∈ U, that



eT E 0 = −

t0

≤

=−

T+



rc (uc (t ), yc (t ))dt −

Z[t

 −

sup

u(· )∈ U, T ≥t0







T

Z[t

rc (uc (t ), yc (t ))dt +

t0



rc (uc (t ), yc (t ))dt −

t0

inf

u(· )∈ U, T ≥t0

T

rd (ud (tk ), yd (tk ))

0 ,T+ )

rd (ud (tk ), yd (tk ))

0 ,T )

 Z[t

 

rd (ud (tk ), yd (tk ))

0 ,T )

q

= Ua (E0 ),

E 0 ∈ R+ .

(41)

Alternatively, it follows from (31), with E (−T− ) = 0 for some −T− ≤ t0 and u(· ) ∈ Ur , that



eT E 0 =

t0

rc (uc (t ), yc (t ))dt +

−T−

≥





rd (ud (tk ), yd (tk ))

Z[−T− ,t ) 0



t0

inf

u(· )∈ Ur , T ≥−t0

−T−

rc (uc (t ), yc (t ))dt +





rd (ud (tk ), yd (tk ))

Z[−T− ,t ) 0

q

= Ur (E0 ),

E 0 ∈ R+ .

(42)

Thus, (41) and (42) imply that (39) is satisfied and q

Ur (E0 ) ≤ eT E0 ≤ Ua (E0 ),

E 0 ∈ R+ .

(43)

Conversely, it follows from (31) and the fact that U (E ) = eT E ≥ 0, E ∈

eT E (t0 ) ≥ −



T

t0

rc (uc (t ), yc (t ))dt −

Z[t

which implies that

e E (t0 ) ≥ T

 sup

u(· )∈ U, T ≥t0

=−





−



T

t0



T

inf

u(· )∈ U, T ≥t0

= Ua (E (t0 )),

t0

rd (ud (tk ), yd (tk )),

q R+ ,

that, for all T ≥ t0 and u(· ) ∈ U, q

E (t0 ) ∈ R+ ,

(44)

0 ,T )

rc (uc (t ), yc (t ))dt −

 Z[t

rc (uc (t ), yc (t ))dt +

 rd (ud (tk ), yd (tk ))

0 ,T )

 Z[t

 rd (ud (tk ), yd (tk ))

0 ,T )

q

E (t0 ) ∈ R+ .

(45) q R+ ,

Furthermore, it follows from the definition of Ua ( · ) that Ua (E ) ≥ 0, E ∈ since the infimum in (37) is taken over the set of values containing the zero value (T = t0 ). q Next, note that it follows from (31), with E (t0 ) ∈ R+ and E (−T ) = 0 for all T ≥ −t0 and u(· ) ∈ Ur , that

eT E (t0 ) =



t0

−T

rc (uc (t ), yc (t ))dt +

 Z[−T,t ) 0

rd (ud (tk ), yd (tk ))

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

 =



t0

inf

u(· )∈ Ur , T ≥−t0

= Ur (E (t0 )),

−T

rc (uc (t ), yc (t ))dt +



rd (ud (tk ), yd (tk ))

Z[−T,t ) 0 q

E (t0 ) ∈ R+ .

(46)

Moreover, since the system G is reachable from the origin, it follows that for every E (t0 ) ∈ u(· ) ∈ Ur such that



t0 −T

13



rc (uc (t ), yc (t ))dt +



q R+ ,

there exists T ≥ −t0 and

rd (ud (tk ), yd (tk ))

(47)

Z[−T,t ) 0

q

is finite, and hence, Ur (E (t0 )) < ∞, E (t0 ) ∈ R+ . Finally, combining (43), (45), and (46), it follows that (40) holds.

 q

It follows from (40) and the definitions of available energy Ua (E0 ) and the required energy supply Ur (E0 ), E0 ∈ R+ , that the hybrid thermodynamic model G can deliver to its surroundings all of its stored subsystem energies and can store all of the work done to all of its subsystems. This is in essence a statement of the hybrid version of the first law of thermodynamics and places no limitation on the possibility of transforming heat into work or work into heat. 5. Nonconservation of entropy and the hybrid second law of thermodynamics In this section, we extend the definition of entropy for the continuous-time thermodynamic systems developed in [1] to give a definition of entropy for the hybrid thermodynamic model (15) and (16) or, equivalently, (23) and (24). To ensure a thermodynamically consistent energy flow model, we require some additional notation and definitions, and additional assumptions to hold for the hybrid large-scale dynamical system G given by (15) and (16). Since our thermodynamic compartmental model involves intercompartmental flows representing energy transfer between compartments, we can use graph-theoretic notions with undirected graph topologies (i.e., bidirectional energy flows) to capture the compartmental system interconnections. Graph theory [26,27] can be useful in the analysis of the connectivity properties of compartmental systems. In particular, a directed graph can be constructed to capture a compartmental model in which the compartments are represented by nodes and the flows are represented by edges or arcs. In this case, the environment must also be considered as an additional node. Specifically, let G(C ) = (V, E ) be a directed graph (or digraph) denoting the thermodynamic network with the set of nodes (or vertices) V = {1, . . . , q} involving a finite nonempty set denoting the subsystems, the set of edges E ⊆ V × V involving a set of ordered pairs denoting the direction of energy flow, and a connectivity matrix C ∈ Rq×q such that C(i, j ) = 1, i, j = 1, . . . , q if ( j, i ) ∈ E, while C(i, j ) = 0 if ( j, i ) ∈ / E. The edge ( j, i ) ∈ E denotes that subsystem j can obtain energy from agent i, but not necessarily vice versa. A graph or undirected graph G(C ) associated with the connectivity matrix C ∈ Rq×q is a directed graph for which the arc set is symmetric, that is, C = C T . Between resettings, the energy flow functions φ cij ( · ), i, j = 1, . . . , q, are assumed to satisfy the following two axioms: Axiom i): For the connectivity matrix C ∈ Rq×q associated with the hybrid large-scale dynamical system G defined by



C(i, j ) =

0, 1,

if

φci j (E (t )) ≡ 0, otherwise,

i = j,

i, j = 1, . . . , q,

t ≥ 0,

(48)

and

C(i,i ) = −

q 

C(k,i ) ,

i = j,

i = 1, . . . , q,

(49)

k=1, k =i

rank C = q − 1, and for C(i, j ) = 1, i = j, φci j (E (t )) = 0 if and only if Ei (t ) = E j (t ) for all E (t ) ∈ / Z, t ≥ 0. Axiom ii): For i, j = 1, . . . , q, [Ei (t ) − E j (t )]φci j (E (t )) ≤ 0, E (t ) ∈ / Z, t ≥ 0. Furthermore, across resettings the energy difference is asuumed to satisfy the following axiom: Axiom iii): For i, j = 1, . . . , q, [Ei (tk+1 ) − E j (tk+1 )][Ei (tk ) − E j (tk )] ≥ 0 for all Ei (tk ) = Ej (tk ), E (tk ) ∈ Z, k ∈ Z+ . The condition φci j (E (t )) = 0 if and only if Ei (t ) = E j (t ), i = j, for all E (t ) ∈ / Z implies that subsystems Gi and G j are connected, and hence, can exchange energy; alternatively φ cij (Ei , Ej ) ≡ 0 implies that subsystems Gi and G j are disconnected, and hence, cannot exchange energy. Axiom i) implies that if the energies in the connected subsystems Gi and G j are equal, then energy exchange between these subsystems is not possible. This statement is consistant to the zeroth law of thermodynamics, which postulates that temperature equality is a necessary and sufficient condition for thermal equilibrium. Furthermore, if C = C T and rank C = q − 1, then it follows that the connectivity matrix C is irreducible, which implies that for any pair of subsystems Gi and G j , i = j, of G there exists a sequence of connectors (arcs) of G that connect Gi and G j . Axiom ii) implies that energy flows from more energetic subsystems to less energetic subsystems and is consistant to the second law of thermodynamics, which states that heat (i.e., energy in transition) must flow in the direction of lower temperatures. Finally, Axiom iii) implies that for any pair of connected subsystems Gi and G j , i = j, the energy difference between consecutive jumps is monotonic.

14

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

Next, we give a hybrid definition of entropy for the hybrid large-scale dynamical system G that generalizes the continuous-time entropy definition established in [1]. q

Definition 5.1. For the hybrid large-scale dynamical system G given by (15) and (16), a function S : R+ → R satisfying

S (E (T )) ≥ S (E (t1 )) +



T

t1

q q    Sci (t ) − σcii (E (t )) Sdi (tk ) − σdii (E (tk )) dt + , c + Ei (t ) c + Ei (tk+ ) i=1

k∈Z[t

1 ,T ]

T ≥ t1 ,

(50)

i=1

where k ∈ Z[t1 ,T )  {k : t1 ≤ tk < T }, c > 0, is called an entropy function of G. The next result gives necessary and sufficient conditions for establishing the existence of a hybrid entropy function of G over an interval t ∈ (tk , tk+1 ] involving the consecutive resetting times tk and tk+1 , k ∈ Z+ . Theorem 5.1. Consider the hybrid large-scale dynamical system G given by (15) and (16), and assume Axioms i)–iii) hold. Then q a function S : R+ → R is an entropy function of G if and only if

S (E (tˆ)) ≥ S (E (t )) +

tˆ  q Sci (t ) − σcii (E (t )) dt, c + Ei (t ) t

tk < t ≤ tˆ ≤ tk+1 ,

(51)

i=1

S (E (tk+ )) ≥ S (E (tk )) +

q  Sdi (tk ) − σdii (E (tk )) , c + Ei (tk+ )

k ∈ Z+ .

(52)

i=1

Proof. Let k ∈ Z+ and suppose S (E ) is an entropy function of G. Then (50) holds. Now, since for tk < t ≤ tˆ ≤ tk+1 , Z[t,tˆ) = ∅, it follows that (51) is immediate. Next, note that

S(E (tk+ )) ≥ S(E (tk )) + which, since Z[t

+ k ,tk )



tk+

tk

q q   Sci (t ) − σcii (E (t )) Sdi (tk ) − σdii (E (tk )) dt + , c + Ei (t ) c + Ei (tk+ ) i=1

(53)

i=1

= k, implies (52).

Conversely, suppose (51) and (52) hold, and let tˆ ≥ t ≥ t1 and Z[t,tˆ) = {i, i + 1, . . . , j}. If Z[t,tˆ) = ∅, then it follows from (51) that S (E ) is an entropy function G. Alternatively, if Z[t,tˆ) = ∅, it follows from (51) and (52) that

S (E (tˆ)) − S (E (t )) = S (E (tˆ)) − S (E (t + )) + j

j−i−1



m=0

[S (E (t + )) − S (E (t +j−m−1 ))] j−m

+ S (E (ti+ )) − S (E (t )) = S (E (tˆ)) − S (E (t + )) + j

j−i 

[S (E (t + )) − S (E (t j−m )] j−m

m=0 j−i−1

+



m=0

[S (E (t j−m )) − S (E (t + ))] + S (E (ti )) − S (E (t )) j−m−1

tˆ  q j−i q   Sdi (t j−m ) − σdii (E (t j−m )) Sci (t ) − σcii (E (t )) ≥ dt + + c + Ei (t ) c + Ei (t + ) tj j−m m=0 i=1

i=1

+

j−i−1 t  j−m m=0

ti  q q  Sci (t ) − σcii (E (t )) Sci (t ) − σcii (E (t )) dt + dt c + E ( t ) c + Ei (t ) t+ t i j−m−1 i=1

i=1

tˆ  q q   Sci (t ) − σcii (E (t )) Sdi (tk ) − σdii (E (tk )) = dt + , c + E ( t ) c + Ei (tk+ ) t i i=1

which implies that S(E) is an entropy function of G.

(54)

k∈Z[t,tˆ) i=1



The next theorem establishes the existence of a continuously differentiable entropy function for the hybrid large-scale dynamical system G given by (15) and (16). Theorem 5.2. Consider the hybrid large-scale dynamical system G given by (15) and (16), and assume Axioms ii) and iii) hold. q Then the function S : R+ → R given by

S (E ) = eT loge (ce + E ) − q loge c,

(55)

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

15 T

where loge (ce + E ) denotes the vector natural logarithm given by [loge (c + E1 ), . . . , loge (c + Eq )] and c > 0, is a continuously differentiable entropy function of G. In addition,

S˙ (E (t )) ≥

q  Sci (t ) − σcii (E (t )) , c + Ei (t )

E (t ) ∈ Z,

tk < t < tk+1 ,

(56)

i=1

S (E (tk )) ≥

q  Sci (tk ) − σcii (E (tk )) , c + Ei (tk+ )

E (tk ) ∈ Z,

k ∈ Z+ .

(57)

i=1

Proof. Since, by Proposition 2.1, E(t) ≥ ≥ 0, E (t ) ∈ Z, t ∈ (tk , tk+1 ], k ∈ Z+ , and φci j (E ) = −φc ji (E ), i = j, i, j = 1, . . . , q, it follows that

S˙ (E (t )) =

q  i=1

=

E˙ i (t ) c + Ei (t )

q  Sci (t ) − σcii (E (t ))

c + Ei (t )

i=1

+

φci j (E (t )) c + Ei (t ) j=1, j =i q 

q   Sci (t ) − σcii (E (t ))  φci j (E (t )) φci j (E (t ))  = + − c + Ei (t ) c + Ei (t ) c + E j (t ) i=1 j=i+1 q

=

q q−1 q  Sci (t ) − σcii (E (t ))   + c + Ei (t ) i=1

i=1

 Sci (t ) − σcii (E (t )) , c + Ei (t )

φci j (E (t ))[E j (t ) − Ei (t )] (c + Ei (t ))(c + E j (t )) j=i+1

q

≥

E (t ) ∈ Z,

tk < t ≤ tk+1 .

(58)

i=1

Furthermore, since E (tk ) ≥≥ 0, E (tk ) ∈ Z, k ∈ Z+ , and φdi j (E ) = −φd ji (E ), i = j, i, j = 1, . . . , q, it follows that

S (E (tk )) =

=

 Ei (tk ) c + Ei (tk ) i=1



−1 q  Ei (tk ) Ei (tk ) ≥ 1+ c + Ei (tk ) c + Ei (tk ) i=1 q 

q  i=1



loge 1 +

Ei (tk ) c + Ei (tk ) + Ei (tk )

q  Ei (tk ) = c + Ei (tk+ ) i=1

=

q 



i=1

φdi j (E (tk )) c + Ei (tk+ ) i=1, j =i

q  Sdi (tk ) − σdii (E (tk )) + c + Ei (tk+ )



 φ (E (t )) φ (E (t ))  di j di j k k − c + Ei (tk+ ) c + E j (tk+ ) j=i+1

q q−1 q  Sdi (tk ) − σdii (E (tk ))   = + + c + Ei (tk ) i=1

i=1

q−1

=

φdi j (E (tk ))[E j (tk+ ) − Ei (tk+ )] (c + Ei (tk+ ))(c + E j (tk+ )) j=i+1

q  Sdi (tk ) − σdii (E (tk ))   + + c + Ei (tk ) q

i=1

i=1

 Sdi (tk ) − σdii (E (tk )) , c + Ei (tk+ ) q

≥

(59)

i=1

where in (59) we use the fact that Theorem 5.1. 

x 1+x

< loge (1 + x ) < x, x > −1, x = 0. The result is now an immediate consequence of

Note that it follows from Theorem 5.1, (58), and (59) that the entropy function given by (55) satisfies (50) as an equality for an equilibrium process and as a strict inequality for a nonequilibrium process. The entropy expression given by (55) is identical in form to the Boltzmann entropy for statistical thermodynamics. Due to the fact that the entropy is indeterminate to the extent of an additive constant, we can place the constant qloge c to zero by taking c = 1. Since S (E ) given by

16

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

(55) achieves a maximum when all the subsystem energies Ei , i = 1, . . . , q, are equal, entropy can be thought of as a measure of the tendency of a system to lose the ability to do useful work, lose order, and settle to a more homogeneous state [1,20]. 6. Semistability and energy equipartition of hybrid thermodynamic systems Inequality (50) is a generalization of Clausius’ inequality for equilibrium and nonequilibrium thermodynamics as well as reversible and irreversible thermodynamics as applied to hybrid large-scale thermodynamic systems involving discontinuous phase transitions. For the (adiabatically) isolated hybrid large-scale dynamical system G, it follows from Theorem 5.1 that (50) yields the hybrid fundamental inequalities

S (E (tˆ)) ≥ S (E (t )),

E (t ) ∈ / Z,

S (E (tk+ )) ≥ S (E (tk )),

tk < t ≤ tˆ ≤ tk+1 ,

E (tk ) ∈ Z,

(60)

k ∈ Z+ .

(61)

Inequalities (60) and (61) imply that, for any dynamical change in an isolated hybrid large-scale thermodynamic system, the entropy between resetting events as well as across resetting events is monotonically nondecreasing. The above observations imply that when an isolated hybrid large-scale dynamical system (i.e., Sc (t) ≡ 0, σ cii (E(t)) ≡ 0, Sd (tk ) ≡ 0, and σ dii (E(tk )) ≡ 0) with thermodynamically consistent energy flow characteristics (i.e., Axioms i)–iii) hold) is at a state of maximum entropy consistent with its energy, it cannot be subject to any further dynamical change since any such change would result in a decrease of entropy. This of course implies that the state of maximum entropy is the stable state of an isolated hybrid thermodynamic system and this state has to be semistable. The next theorem concretizes the above observations. First, however, the following definition for semistability is needed. Recall that for addressing the stability of an impulsive dynamical system the usual stability definitions are valid [14]. Here, q we need to appropriately modify these definitions to restrict initial conditions to the positively invariant subset R+ of Rq . q

q

Definition 6.1. An equilibrium solution x(t ) ≡ xe ∈ R+ to (1) and (2) is semistable with respect to R+ if it is Lyapunov stable with respect to

q R+

and there exists δ > 0 such that if x0 ∈ Bδ (xe ) ∩ q

n R+ ,

then limt → ∞ x(t) exists and corresponds to a Lyaq

punov stable equilibrium point with respect to R+ . The system (1) and (2) is said to be semistable with respect to R+ if every equilibrium point of (1) and (2) is semistable with respect to q

q R+ .

Finally, (1) and (2) is said to be globally semistable with q

respect to R+ if every equilibrium point of (1) and (2) is globally semistable with respect to R+ . Theorem 6.1. Consider the hybrid large-scale dynamical system G given by (15) and (16) with Sc (t) ≡ 0, Sd (t) ≡ 0, σ cii (E(t)) ≡ 0, and σ dii , and assume that Axioms i ) − iii ) hold. Then, for every α ≥ 0, α e is a semistable equilibrium state of (15) and (16). q Furthermore, E (t ) → 1q eeT E (t0 ) as t → ∞ and 1q eeT E (t0 ) is a semistable equilibrium state with respect to R+ . q

Proof. It follows from Axiom i) that α e ∈ R+ , α ≥ 0, is an equilibrium state of (15) and (16). To show Lyapunov stability of the equilibrium state α e, consider the Lyapunov function candidate E (E ) = 12 (E − α e )T (E − α e ). Since φci j (E ) = −φc ji (E ), E ∈ q q R+ , i = j, i, j = 1, . . . , q, and eT fc (E ) = 0, where fci (E ) = j=1, j =i [σci j (E ) − σc ji (E )], i = 1, . . . , q, it follows from Axiom ii) that

E˙ (E (t )) = (E (t ) − α e )T E˙ (t ) = (E (t ) − α e )T fc (E (t )) = E T (t ) fc (E (t )) =

q 



Ei (t )

i=1

=

q 

 φci j (E (t ))

j=1, j =i

q q  

[Ei (t ) − E j (t )]φci j (E (t ))

i=1 j=i+1

≤ 0,

E (t ) ∈ Z.

(62)

Next, it follows from Axioms ii) and iii) that

E (E (tk )) = =

1 1 (E (tk+ ) − α e )T (E (tk+ ) − α e ) − (E (tk ) − α e )T (E (tk ) − α e ) 2 2 q q   i=1 j=1, j =i

=

q−1 q  

q 1 Ei (tk+ )φdi j (E (tk )) − 2



i=1

[Ei (tk+ ) − E j (tk+ )]φdi j (E (tk ))

i=1 j=i+1

q  j=1, j =i

2

φdi j (E (tk ))

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

q 1 − 2



i=1

q 

2 φdi j (E (tk ))

j=1, j =i

E (tk ) ∈ Z,

≤ 0,

(63)

which, by Theorem 2.1, establishes Lyapunov stability of the equilibrium state α e with respect to q

To show that α e is semistable with respect to R+ , note that

E˙ (E (t )) =

q q  

17

q R+ .

[Ei (t ) − E j (t )]φci j (E (t ))

i=1 j=i+1

=

q  

[Ei (t ) − E j (t )]φci j (E (t )),

E (t ) ∈ Z,

(64)

i=1 j∈Ki

and

E (E (tk )) =

q q   i=1 j=1, j =i

≥

q 1 Ei (tk )φdi j (E (tk )) + 2

q−1 q  

i=1



2

q 

φdi j (E (tk ))

j=1, j =i

[Ei (tk ) − E j (tk )]φdi j (E (tk ))

i=1 j=i+1

=

q−1  

[Ei (tk ) − E j (tk )]φdi j (E (tk )),

E (tk ) ∈ Z,

(65)

i=1 j∈Ki

−1 where Ki = Ni \ ∪il=1 {l } and Ni = { j ∈ {1, . . . , q} : φci j (E ) = 0 and φdi j (E ) = 0 if and only if Ei = E j }. It now follows from ˙ (64) that E (E ) = 0 if and only if (Ei − E j )φci j (E ) = 0, E ∈ Z, i = 1, . . . , q, j ∈ Ki . Next, we show that E (E ) = 0 if and only if (Ei − E j )φdi j (E ) = 0, E ∈ Z, i = 1, 2, . . . , q, j ∈ Ki . Assume that (Ei − E j )φdi j (E ) = 0, E ∈ Z, i = 1, . . . , q, j ∈ Ki . Then, it follows from (65) that E (E ) ≥ 0, E ∈ Z, and hence, by (63), E (E ) = 0, E ∈ Z. Conversely, assume E (E ) = 0, E ∈ Z. In this case, it follows from (63) that [Ei (tk+ ) − E j (tk+ )]φdi j (E (tk )) = 0 and q φ (E (tk )) = 0, E (tk ) ∈ Z, i, j = 1, . . . , q, i = j. Now, j =1, j =i di j



Ei (tk+ ) − E j (tk+ )

   φdi j (E (tk )) = Ei (tk ) − E j (tk ) φdi j (E (tk ))  q 

+

φdih (E (tk )) −

h=1,h =i

q 

 φd jl (E (tk )) φdi j (E (tk ))

l =1,l = j

= [Ei (tk ) − E j (tk )]φdi j (E (tk )), E (tk ) ∈ Z, i, j = 1, . . . , q, i = j, and hence, (Ei − E j )φdi j (E ) = 0, E ∈ Z, i = 1, . . . , q, j ∈ Ki . Finally, let



 

q

q

R = E ∈ R+ : E ∈ / Z, E˙ (E ) = 0 ∪ E ∈ R+ : E ∈ Z, E (E ) = 0



= E ∈ Rq+ : E ∈ / Z, (Ei − E j )φci j (E ) = 0, i = 1, . . . , q, j ∈ Ki



(66)







∪ E ∈ Rq+ : E ∈ Z, (Ei − E j )φdi j (E ) = 0, i = 1, . . . , q, j ∈ Ki , q

and note that Axiom i) implies that R = {E ∈ R+ : E1 = · · · = Eq }. Since the set R consists of the equilibrium state of the system, it follows that the largest invariant set M contained in R is given by M = R. Hence, it follows from Theorem 2.2 that q for every initial condition E (t0 ) ∈ R+ , E (t ) → M as t → ∞, and hence, α e is a semistable equilibrium state of the system. T T Next, note that since e E (t ) = e E (t0 ) and E (t ) → M as t → ∞, it follows that E (t ) → 1q eeT E (t0 ) as t → ∞. Hence, with

α = 1q eT E (t0 ), α e = 1q eeT E (t0 ) is a semistable state of the hybrid dynamical system (15) and (16). 

Theorem 6.1 implies that the steady-state value of the energy in each subsystem Gi of the isolated hybrid large-scale dynamical system G is equal, that is, the steady-state energy of the isolated hybrid large-scale dynamical system G given by



E∞

1 = eeT E (t0 ) = q



q 1 Ei (t0 ) e q

(67)

i=1

is uniformly distributed over all subsystems of G. This phenomenon is known as equipartition of energy [1,20,28] and is an emergent behavior in thermodynamic systems.

18

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

1 3

4

2

5

Fig. 7.1. Thermodynamic model with bidirectional heat flow.

7. Resetting set structure and an illustrative numerical example As noted in Section III, the type of phase transition as well as the phase transition driving parameters will determine the structure of the resetting set, whereas the nonequilibrium thermodynamic process will determine the hybrid dynamical equations of motion describing the evolution of the flow of energy or particles through the system. For example, a peritectic transformation, in which a two-component single phase solid is heated and transforms into a solid phase and a liquid phase, and the emergence of superconductivity in certain metals and ceramics when cooled below a critical state will involve different resetting set structures and governing hybrid dynamical equations of motion. However, the generic form of the governing equations will involve an underlying hybrid thermodynamic model as developed in Section III. For illustrative purposes, in this section we consider a generic thermodynamic structure for the resetting set Z. Specifically, let Oi denote the set of all compartments with energy (heat) flowing out to the ith compartment, let Ii denote the set of all compartments receiving energy (heat) from the ith compartment, and let V = {1, . . . , q} denote the set of vertices representing the intercompartmental connections of the hybrid compartmental thermodynamic system. Furthermore, define the local resetting set Zi by



Zi 

E i ∈ R+ :



φci j (Ei , E j )(Ei − E j ) −

j∈Oi





φci j (Ei , E j )(Ei − E j ) = 0 and Ei = E j ,

j ∈ Oi ∪ Ii ,

i = 1, . . . , q,

j∈Ii

(68) with

Z

q 

q

{ E ∈ R+ : E i ∈ Z i } .

(69)

i=1

It follows from (58) that (68) implies that if the time rate of change of the difference in entropies in the connected input flow and output flow subsystems in zero and energy equipartition is not reached, then a resetting occurs. For our hybrid large-scale thermodynamic system G a straightforward, but lengthy, calculation shows that Assumptions A1, A2, and 2.2 hold. For a similar calculation as well as additional resetting set structures, consistent with thermodynamic principles, see [14,29]. Next, we consider the generic five-compartment thermodynamic system shown in Fig. 7.1. Here, we consider continuous and discrete energy flows from the ith subsystem to all other neighboring subsystems given by the sum of the individual energy flows from the ith subsystem to the jth subsystem. Furthermore, we assume that the energy flows are proportional to the energy differences of the subsystems, that is, Ei − E j . This gives a thermodynamic model that is completely analogous to the equations of thermal transfer with subsystem energies playing the role of temperatures. Now, a power balance that governs the energy exchange among the coupled subsystems gives

E˙ 1 (t ) =

1 [E2 (t ) − E1 (t )], 5

E˙ 2 (t ) =

1 [E1 (t ) − E2 (t ) + E3 (t ) − E2 (t ) + E5 (t ) − E2 (t )], 5

E˙ 3 (t ) =

1 [E2 (t ) − E3 (t ) + E4 (t ) − E3 (t )], 5

E˙ 4 (t ) =

1 [E3 (t ) − E4 (t )], 5

E4 (0 )=E40 ,

E4 (t ) ∈ / Z4 ,

(73)

E˙ 5 (t ) =

1 [E2 (t ) − E5 (t )], 5

E5 (0 )=E50 ,

E5 (t ) ∈ / Z5 ,

(74)

E1 (0 )=E10 ,

E1 (t ) ∈ / Z1 ,

E3 (0 )=E30 ,

(70) E2 (0 )=E20 , E2 (t ) ∈ / Z2 , E3 (t ) ∈ / Z3 ,

(71)

(72)

whereas an energy balance that governs the energy resetting among all the subsystems is given by

1 5

E1 (t ) = [E2 (t ) − E1 (t )],

E1 (t ) ∈ Z1 ,

(75)

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

19

Fig. 7.2. System energies versus time; E1 (t) in blue, E2 (t) in red, E3 (t) in green, E4 (t) in magenta, and E5 (t) in black. (See color figure online.). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7.3. System entropy versus time.

1 5

E2 (t ) = [E1 (t ) − E2 (t ) + E3 (t ) − E2 (t ) + E5 (t ) − E2 (t )], 1 5

E3 (t ) = [E2 (t ) − E3 (t ) + E4 (t ) − E3 (t )],

E3 (t ) ∈ Z3 ,

E2 (t ) ∈ Z2 ,

(76) (77)

1 5

E4 (t ) ∈ Z4 ,

(78)

1 5

E5 (t ) ∈ Z5 ,

(79)

E4 (t ) = [E3 (t ) − E4 (t )], E5 (t ) = [E2 (t ) − E5 (t )],

20

W.M. Haddad / Commun Nonlinear Sci Numer Simulat 83 (2020) 105096

where 1/5 is the system thermal conductivity. 5

It follows form Theorem 6.1 that the hybrid thermodynamic heat flow model (70)–(79) is semistable with respect to R+ and achieves energy equipartition. To see this, let E10 = 0, E20 = 10, E30 = 20, E40 = 30, and E50 = 40. Fig. 7.2 shows the energy trajectories of each compartment versus time. The entropy profile as a function of time is plotted in Fig. 7.3. 8. Conclusion In this paper, we combined classical thermodynamic notions with hybrid dynamical systems theory to provide a systemtheoretic foundation for hybrid thermodynamics. The proposed framework captures jump discontinuities in the fundamental thermodynamic state quantities by developing a hybrid large-scale dynamical system using impulsive compartmental and thermodynamic dynamical system models involving an interacting mixture of continuous and discrete dynamics exhibiting discontinuous flows on appropriate manifolds. In future research, we will extend our hybrid large-scale dynamical system model to consider work done by the system on the environment as well as work done by the environment on the system. This extension can be addressed by including an additional state equation, coupled to the hybrid energy balance equation, involving volume (deformation) states for each subsystem [20]. Declaration of Competing Interest The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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