Region stability of switched two-dimensional linear dissipative Hamiltonian systems with multiple equilibria

2.1. System description and preliminaries

Consider a switched linear Hamiltonian system with multiple equilibrium subsystems as follows.

where $$ x=[x_{1}, x_{2}]^T\in\mathbb{D} $$ represents the state of the system, and $$ \mathbb{D} $$ is the common domain of all the subsystems of system (1). The map $$ \sigma(t)\colon[t_{0}, +\infty)\to\Lambda:=\{1, 2\} $$ is a piecewise right continuous constant step function, which is called to be a switching path or a switching rule. The value of the function $$ \sigma(\mathrm{t})=i $$, where $$ i=1, 2 $$, means that the subsystem $$ {i} $$ is activated at the time instant $$ t $$. The matrices $$ J_{i}\in\mathbb{R}^{2\times2} $$ and $$ R_i\in\mathbb{R}^{2\times2} $$ are all constant matrices. Moreover, $$ J_{i}^{T}=-J_{i}\not= 0 $$ and $$ R_{i}^{T}=R_{i} $$, $$ i=1, 2 $$. The notation $$ \nabla H_{i}(x)=\frac{\partial H_{i}(x)}{\partial x} $$ is the gradient of the energy function $$ H_i(x) $$ of subsystem $$ i $$. The function $$ H_i(x) $$, where $$ i=1, 2 $$, satisfies the following two: (1) they are continuous and differentiable; (2) $$ H_i(x)>0 $$ for any $$ x\in \mathbb{D}-\{x^{e1}, x^{e2}\} $$, in which $$ x^{ei} $$ satisfying $$ H_{i}(x^{ei})=0 $$ is a unique equilibrium point of subsystem $$ {i} $$. Moreover, the equilibrium points $$ x^{e1} $$ and $$ x^{e2} $$ of the two subsystems are different. In fact, system (1) considered in this paper is a switched two-dimensional linear system only consisting of two subsystems.

Remark 1Note that due to the special dissipative Hamiltonian structures of subsystems of system (1), i.e., the matrices $$ J_i^T=-J_i $$ and $$ R_i^T=R_i{\geqslant}0 $$, if the matrix $$ J_i\not=0 $$, then the directions of trajectories of subsystems of system (1) are all revolved around the corresponding equilibrium points of subsystems anticlockwise or clockwise.

2.2. Some definitions and propositions

This subsection refers to some definitions and gives a proposition that is needed to analyze the region stability of system (1) in the next section below.

Definition 1 (The maximum energy function) ^[15]. The following function $$ H(x) $$ is called the maximum energy function of system (1)

Definition 2(The maximum switching energy sequence) ^[15]. The following sequence is known as the maximum switching energy sequence of the switching path $$ \sigma(t) $$

Proposition 1Consider system (1). The Hamiltonian functions $$ H_{1}(x) $$ and $$ H_{2}(x) $$ of the subsystems satisfy the following formula

where $$ \lambda_{\min}(Q_i) $$ and $$ \lambda_{\max}(Q_i) $$ are the minimum and maximum eigenvalues, respectively, of the matrix $$ Q_i $$ for $$ i=1, \, 2 $$.

Proof: Letting

We obtain from the Hamiltonian function $$ H_i(x) $$ of the $$ i $$-th subsystem expressed as in (5) that

Since every $$ Q_{i} $$ is a real symmetric positive definite matrix, there is an orthogonal matrix $$ P_{i} $$ satisfying

where $$ E_i $$ is an identity matrix, and an orthogonal transformation

such that the following two hold.

(A.) By means of the orthogonal matrix $$ P_{i} $$, the positive definite matrix $$ Q_i $$ can be diagonalized into a diagonal matrix as follows.

where $$ \lambda_{i1} $$ and $$ \lambda_{i2} $$ are the two real positive eigenvalues of the positive definite matrix $$ Q_{i} $$, where $$ i=1, \, 2 $$.

(B.) The Hamiltonian function $$ H_{i}(x^{i}) $$ in (14) can be transformed by the orthogonal transformation (16) into the following standard form:

where $$ y^i $$ is as follows.

It can be obtained from (13), (19), (16), and (15) that the square norm of the vector $$ x^{i} $$ is as follow.

We know from (17)-(20) that the following two hold.

and

where $$ i=1, \, 2 $$.

It follows from (14), (18), (21), and (22) that

Therefore, we know from (17), (20), and (23) that for any $$ x\in\mathbb{D}-\{x^{ei}\} $$, the following holds.

which is exact (12). The proof of Proposition 1 is thus completed. □

Since every subsystem of system (1) has a unique equilibrium point $$ x^{ei} $$, $$ \; $$$$ i=1, 2 $$, and the maximum energy functions in Definition 1 are continuous everywhere in $$ \mathbb{D} $$; there exists a unique compact region that is defined as

or

Similar to that of region stability defined in the reference ^[17], based on the region $$ \Omega $$ in (24) or the region $$ \Psi $$ in (25), we introduce the concept of region stability for system (1) as follows.

Definition 3^[17] Consider system (1) with the region $$ {\Omega} $$ defined in (24) or$$ \Psi $$defined in (25) under a special kind of switching path $$ \sigma(t) $$, i.e., the switching line $$ l_1 $$ in (2). System (1) under the switching line $$ l_1 $$ in (2) is said to be

● Region stable with respect to the region $$ \Omega $$ in (24) or the region $$ \Psi $$ in (25), if for $$ \forall\varepsilon>0, \;\exists\delta:=\delta(\varepsilon)>0 $$ such that the following formula holds for any $$ x_{0} $$,

or

● Asymptotically region stable with respect to the region $$ \Omega $$ in (24) or the region $$ \Psi $$ in (25), if both (26) or (27) and the following limit hold

3.1. Some lemmas

This subsection introduces some lemmas that will be used in the next subsection. Firstly, it can be obtained from Proposition 1 that the following result holds.

Lemma 1Consider system (1) under the switching line (2). The Hamiltonian functions $$ H_{1}(x) $$ and $$ H_{2}(x) $$ and the maximum energy function $$ H(x) $$ satisfy the following inequation.

where $$ \alpha=\min\big\{\lambda_{\min}(Q_1), \lambda_{\min}(Q_2)\big\} $$, $$ \beta=\max\big\{\lambda_{\max}(Q_1), \lambda_{\max}(Q_2)\big\} $$, and the region $$ \Omega $$ and the maximum energy function $$ H(x) $$ are defined in (24) and (10), respectively.

Proof: It is easy to see from the maximum energy function defined in (10) of Definition 1 and the equation (12) in Proposition 1 and Condition (24) that Lemma 1 holds true. □

Lemma 2Consider system (1) under the switching line (2). If $$ R_{i}{\geqslant} 0\text{} $$, then for the trajectory $$ x(t)\in\mathbb{D}-\Omega $$, $$ \forall t\in[t_m, \, t_{m+1}) $$, the two Hamiltonian functions $$ H_{1}\left(x\right) $$ and $$ H_{2}\left(x\right) $$ have the same variant trend of the properties of either monotonous increase or monotonous decrease at all the switching states $$ x_m $$, where any $$ m\in\mathbb{N} $$. That is, for all the switching states $$ x_{m}, x_{m+1}\in\mathbb{D}-\Omega $$, $$ \bigtriangleup H_1:=H_1(x_{m})-H_1(x_{m+1}) $$ and $$ \triangle H_2:=H_2(x_{m})-H_2(x_{m+1}) $$ satisfy

where $$ \begin{matrix}{\rm sign}(\cdot)\\ \end{matrix} $$ denotes the sign function.

Proof: Since for $$ x(t)\in\mathbb{D}-\Omega $$, $$ \forall t\in[t_m, t_{m+1}) $$, all the switching states $$ x_m $$ are situated in the switching line $$ l_{1} $$ in (2), the following formula holds

where $$ x_{1, m} $$ and $$ x_1^{ei_m} $$ denote the first elements of the switching state $$ x_m $$ and the equilibrium point $$ x^{ei_m} $$, respectively; And the parameters $$ g_{i_{m}} $$ are defined as follows.

It follows from the fact that the trajectory $$ x(t)\in\mathbb{D}-\Omega $$ and $$ R_{i_{m}}(x){\geqslant} 0 $$ that for $$ t\in[t_{m}, t_{m+1}) $$, $$ i_{m}=1, 2 $$, $$ m\in\mathbb{N} $$, the Hamiltonian function $$ H_{i_{m}}(x) $$ satisfies

One obtains from (32) that

Since the first element $$ x_{1}^{ei_{m}} $$ of the equilibrium point $$ x^{ei_{m}} $$ is contained in the region $$ \Omega $$, we know from (24), (29), (31), and (33) that for any $$ m\in\mathbb{N} $$, $$ i_m=1, 2 $$,

It is then obtained from (31) and (34) that for any $$ i_m\in\{1, \, 2\} $$,

One obtains from (31) and (35) that for all $$ i_{m}, \;i_{s}\in\Lambda $$ and $$ i_{m}\neq i_{s} $$, the following holds.

From (35) and (36), it follows that

which is (30). Thus, Lemma 2 holds true.□

Lemma 3^[27] Let

and

If the two infinite sequences $$ \left\{a_{n}^{i}\right\}_{n=1}^{+\infty} $$ and $$ i=1, 2 $$ are both monotonically decreasing/increasing, then $$ \left\{a_{n}^{\min}\right\}_{n=1}^{+\infty} $$ and $$ \{a_{n}^{\max}\}_{n=1}^{+\infty} $$ are also monotonically decreasing/increasing sequences.

Lemma 4^[26] System (1) under the switching line $$ l_1 $$ in (2) is region stable with respect to the region $$ \Omega $$ in (24) if and only if for any $$ x_{0}\in\mathbb{D} $$, the following holds:

where the $$ i_{m}=\sigma(t_{m})\in\{1, 2\} $$; The parameter $$ C $$ is a constant; And $$ H(x) $$ is defined as in Definition 1.

Remark 5Note that the proofs of Lemmas 1-4 are just related to the compact property of the regions of $$ \Omega $$ in (24) and $$ \Psi $$ in (25) containing all the equilibrium points of the subsystems. Therefore, if the region $$ \Omega $$ in (24) is replaced by the region $$ \Psi $$ in (25), then all Lemmas 1-4 hold too.

3.2. Regional stability results

Based on Definition 3, we obtain from Proposition 1, Lemmas 1-4 that the two main results of this paper are proposed in series as follows.

For the horizontal and vertical ordinates of the equilibrium points $$ x^{e1} $$ and $$ x^{e2} $$ of the two subsystems, there are the following two cases: (1) $$ x_{1}^{e1}\neq x_{1}^{e2} $$; (2) $$ x_{1}^{e1}= x_{1}^{e2} $$ and $$ x_{2}^{e1}\neq x_{2}^{e2} $$. For the former, we propose the following region stability result of system (1) under the switching line $$ l_1 $$ in (2) as follows.

Theorem 1Consider system (1) with the compact region $$ \Omega $$ in (24) and the switching line $$ l_1 $$ in (2). For the case that $$ x_{1}^{e1}\neq x_{1}^{e2} $$, system (1) under the switching line $$ l_1 $$ in (2) is

(i) region stable with respect to the region $$ \Omega $$ in (24), if $$ J_1\not= 0 $$, $$ J_2\not=0 $$, $$ R_1{\geqslant} 0 $$, $$ R_2{\geqslant} 0 $$, and

where $$ \alpha=\min\big\{\lambda_{\min}(Q_1), \lambda_{\min}(Q_2)\big\} $$, $$ \beta=\max\big\{\lambda_{\max}(Q_1), \lambda_{\max}(Q_2)\big\} $$, and $$ v_{1} $$ is the intersection point of the two parabolic curves $$ L_{1}(x_{1}) $$ and $$ L_{2}(x_{1}) $$ in (6) over the interval $$ (x_{1}^{e1}, \, x_{1}^{e2}) $$ in the horizontal axis, i.e.,

where $$ g_1 $$ and $$ g_2 $$ are defined in (7).

(ii) asymptotic region stable with respect to the region $$ \Omega $$, if $$ J_1\not= 0 $$, $$ J_2\not=0 $$, $$ R_1> 0 $$, $$ R_2> 0 $$, and the condition (41) are all satisfied.

Proof: Without loss of generality, we just show that the two Statements $$ (i) $$ and $$ (ii) $$ hold for the case that $$ x_1^{e1}<x_1^{e2} $$. As for the case that $$ x_1^{e1}>x_1^{e2} $$, it is similar to show that the two Statements also hold true.

For any trajectory $$ x(t) $$ of system (1) under the switching line $$ l_1 $$ in (2) starting from any initial state $$ x_0 $$ at the initial time $$ t_{0} $$, there are the following three cases that should be considered: (a) The trajectory of the system $$ x(t)\in\mathbb{D}-\Omega $$, $$ \forall t>t_{0} $$; (b) The trajectory of the system $$ x(t) $$, $$ \forall t>t_0 $$, is always contained in $$ \Omega $$; (c) The trajectory of the system $$ x(t) $$, $$ \forall t>t_0 $$, is neither always contained in $$ \mathbb{D}-\Omega $$ nor always contained in $$ \Omega $$.

(1) We show the conclusion of Theorem 1 holds for the case that $$ g_1\neq g_2 $$.

Firstly, we will show that the Statement $$ (i) $$ holds for Case (a). Since the Hamiltonian functions $$ H_{1}(x) $$ and $$ H_{2}(x) $$ of the two subsystems are both continuous everywhere in $$ \mathbb{D} $$, the maximum energy function $$ H(x) $$ defined in Definition 1 is also continuous everywhere in $$ \mathbb{D} $$. We obtain from the conditions of $$ Q_i>0 $$, $$ R_i{\geqslant}0 $$, where $$ i=1, \, 2 $$, and the ordinary differential equations (ODEs) (1) that for any $$ t\in[t_m, t_{m+1}) $$ and $$ \text{}m\in\mathbb{N} $$, the following holds.

It follows from (43) that

which can also be expressed as

Since $$ R_i{\geqslant}0 $$ holds for $$ i=1, \, 2 $$, Lemma 2 holds for system (1). Then one knows from (30) in Lemma 2 and (45) that for $$ i_{s}\neq i_{m}\in\{1, \, 2\} $$,

It can be obtained from (30) in Lemma 2, (45), and (46) that the two sequences $$ \big\{H_{1}(x_{m})\big\}_{m=0}^{+\infty} $$ and $$ \big\{H_{2}(x_{m})\big\}_{m=0}^{+\infty} $$ are both monotonically decreasing. It then can be obtained from Lemma 3 that the switching maximum energy sequence $$ \big\{H(x_m)\big\}_{m=0}^{+\infty} $$ is also monotonically decreasing.

Based on the above analysis, one obtains from (10) in Definition 1 and the fact that the trajectory $$ x(t)\in\mathbb{D}-\Omega $$, for all $$ t\in[t_{m}, t_{m+1}) $$, $$ i_m=1, 2 $$, and any $$ m\in\mathbb{N} $$ that

It is easy to see from (47) that

From which and (41), we know that

It follows from (49) that (40) in Lemma 4 is satisfied for system (1). Then, by Lemma 4, we know that system (1) under the switching line $$ l_1 $$ in (2) is region stable with respect to the region $$ \Omega $$ for Case (a).

Secondly, we will show that Statement $$ (i) $$ also holds for case (b). For $$ g_{1} $$ and $$ g_{2} $$ in (2), there are the following two relationships: $$ g_{1}=g_{2} $$ and $$ g_{1}\neq g_{2} $$. We first consider the case that $$ g_{1}\neq g_{2} $$. In this case, letting $$ L_1(x_{1})=L_2(x_{1}) $$ one obtains from (6) that

From which, it can be known that the discriminant of the roots of the equation (50) is as follows.

which means that two curves $$ L_1(x_{1}) $$ and $$ L_2(x_{1}) $$ have two intersection points.

It follows from (51) that the equation (50) has the following two solutions:

and

where $$ v_{1} $$ and $$ v_{2} $$ denote the two intersection points of the two curves $$ L_1(x_{1}) $$ and $$ L_2(x_{1}) $$ satisfying $$ v_{1}\in(x_{1}^{e1}, x_{1}^{e2}) $$ and $$ v_{2}\in(-\infty, x_{1}^{e1})\cup(x_{1}^{e2}, +\infty) $$, respectively.

Next, we will find the intersection points of the switching line $$ l_{1} $$ passing through the boundary of the region $$ \Omega $$. To do that, we obtain from $$ L_1(x_{1})=L_1(x_{1}^{e2}) $$ that

It can be obtained from (54) that the discriminant of the roots of the equation (54) is as follows.

It then follows from (55) that the solutions of the equation (54) are as follows.

The two intersection points can be denoted by $$ p_{1}=x_{1}^{e2} $$ and $$ p_{2}=2x_{1}^{e1}-x_{1}^{e2} $$.

Similarly, solving $$ L_{2}(y_{1})=L_{2}(x_{1}^{e1}) $$ yields that

The two intersections are then denoted by $$ p_{3}= x_{1}^{e1} $$ and $$ p_{4}=2x_{1}^{e2}-x_{1}^{e1} $$, respectively.

One obtains from (56) and (57) that the switching line $$ l_{1} $$ in (2) passes through the largest point and the smallest point of the boundary of the region $$ \Omega $$. The two maximum and minimum points are, respectively, denoted by

In the following, we consider the minimum value of the maximum energy function $$ H(x) $$ over the region $$ \Omega $$.

(C1) As $$ v_{2}\in(-\infty, x_{1}^{e1}) $$, the maximum energy function $$ H(x) $$ is as follows.

From (59), one obtains that the $$ L_2(x) $$ is continuous over the interval $$ (v_{2}, v_{1}]\text{} $$ and $$ {dL_{2}(x_1)\over dx_1}<0 $$, so the minimum value of $$ L_2(x_{1}) $$ over the interval $$ (v_{2}, v_{1}]\text{} $$ is $$ L_{2}(v_{1}) $$.

Similarly, we know from (59) that the $$ L_1(x_{1}) $$ is continuous over the interval $$ \left(v_{1}, +\infty\right) $$ and $$ {dL_{1}(x_1)\over dx_1}>0 $$, so the minimum value of $$ L_1(x_{1}) $$ over the interval $$ \left(v_{1}, +\infty\right) $$ is $$ L_{1}(v_{1}) $$.

It is obvious from (50) and (52) that

From (59) and (60), it shows that the minimum value of $$ H(x) $$ under the case (C1) is as follows.

where $$ v_1 $$ is expressed in (52).

(C2) As $$ v_{2}\in\left(x_{1}^{e2}, +\infty\right) $$, the maximum energy function $$ H(x) $$ is as follows.

It can be obtained from (62) that the $$ L_2(x_{1}) $$ is continuous over the interval $$ \left(-\infty, v_{1}\right) $$ and $$ {d{L_{2}}(x_{1})\over dx_1}<0 $$. Then, the minimum value of the maximum energy function $$ H(x)=L_2(x_{1}) $$ on the interval $$ \left(-\infty, v_{1}\right] $$ is $$ L_{2}(v_{1}) $$.

Similarly, we know that the $$ L_1(x_{1}) $$ is continuous over the interval $$ \left[v_{1}, v_{2}\right] $$ and $$ {d{L_{1}}(x_{1})\over dx_1}>0 $$. Then, the minimum value of the maximum energy function $$ H(x)= L_1(x_{1}) $$ over the interval $$ \left[v_{1}, v_{2}\right] $$ is $$ L_{1}(v_{1}) $$.

Therefore, it can be obtained from (59) and (60) that the minimum value of the maximum energy function $$ H(x) $$ under the case (C2) is as follows.

Then, one obtains from (61) and (63) that the minimum value of the maximum energy function $$ H(x) $$ over the region $$ \Omega $$ is as follows.

It follows from (41) and the condition that $$ R_1>0 $$ and $$ R_2>0 $$ that

and

where $$ { } p_{\max} $$ and $$ { } p_{\min} $$ are the same as in (58).

From (59), (60), (62), (65), and (66), one obtains that

where $$ \max\big\{L_1(p_{\max}), \, L_2(p_{\min})\big\} $$ denotes the maximum value of $$ H(x) $$ over the region $$ \Omega $$.

It is obvious from (64) and (67) that for any switching states $$ x_m $$, the following holds.

which implies that (40) in Lemma 4 is satisfied. By Lemma 4, we know that for Case (b), system (1) under the switching line $$ l_1 $$ in (2) is region stable with respect to the region $$ \Omega $$.

Thirdly, we show that Statement $$ (i) $$ also holds for Case (c) as follows. In such case, the trajectory $$ x(t) $$ of system (1) is not always in the region $$ \mathbb{D}-\Omega $$. For any small enough $$ \varepsilon>0 $$, there is a time interval sequence $$ \left\{\left[t_{m_{\tau}}, t_{m_{\tau}+1}\right]\right\}_{\tau=1}^{+\infty} $$, such that the states $$ x_{m_{\tau}}=x(t_{m_{\tau}};t_0, x_0, i_{m_{\tau}}) $$ and $$ x_{m_{{\tau}}+1}=x(t_{m_{\tau}+1};t_{0}, x_{0}, i_{m_{\tau}+1}) $$ are all on the boundary of the set $$ \Omega_{\varepsilon}:=\big\{z\in\mathbb{D}\colon\mathrm{d}(z, \Omega)\leq\varepsilon\big\} $$. We insert $$ x_{m_{\tau}} $$ and $$ x_{m_{\tau}+1} $$ into the switching state sequence $$ \big\{x_{m}\big\}_{m=0}^{+\infty} $$. Correspondingly, the times $$ t_{m_{\tau}} $$ and $$ t_{m_{\tau}+1} $$ are inserted into the switching time sequence $$ \big\{t_{m}\big\}_{\mathrm{m}=0}^{+\infty} $$, and the indexes $$ i_{m_{\tau}} $$ and $$ i_{m_{\tau}\text{+}1} $$ are all inserted into the switching index sequence $$ \big\{i_{m}\big\}_{\mathrm{m}=0}^{+\infty} $$.

Based on the above statements, we know that there is a time interval $$ \left[t_{m_{\tau}}, t_{m_{\tau}+1}\right] $$ for $$ m_{\tau}\in\mathbb{N} $$, such that for $$ {\tau}\in\mathbb{N} $$, $$ i_{m_{\tau}}, i_{m_{\tau}+1}=1, \, 2 $$,

It is known from (69) that Statement $$ (i) $$ is true for the trajectory of $$ x(t\colon t_{m_{{\tau}}}, x_{m_{{\tau}}}, i_{m_{{\tau}}}) $$ of system (1) contained in $$ \mathbb{D}-\operatorname{int}(\Omega_{\varepsilon}) $$. Thus the proof is similar to that proof of Case (a). On the other hand, it is obvious that as $$ \varepsilon\rightarrow0 $$, The state trajectory $$ x(t) $$ contained in the region $$ \Omega_{\varepsilon}-\Omega $$ will gradually go into the region $$ \Omega $$. Based on the above, we know that Statement $$ (i) $$ holds too.

Finally, Statement $$ (ii) $$ will be shown under the following two Situations.

(S1) As $$ v_{2}\in(-\infty, x_{1}^{e1}) $$ and $$ x(t)\in\mathbb{D}-\Omega $$, for all $$ t\in[t_{m}, t_{m+1}) $$ and any $$ m\in\mathbb{N} $$, the values of the maximum energy function $$ H(x) $$ at the switching states $$ x_m=[x_{1, \, m}\; x_{2, \, m}]^T\in\mathbb{R}^2 $$ and $$ m\in\mathbb{N} $$ are as follows.

The subsystem $$ i_{m} $$ is activating when any $$ t\in[t_{m}, t_{m+1}) $$ and $$ x(t)\in\mathbb{D}-\Omega $$. And it is obvious from Condition $$ R_1>0 $$ and $$ R_2>0 $$ of Statement $$ (ii) $$ that for any $$ i_m=1, \, 2 $$,

We know from (71) that

From Lemma 2, the condition of $$ R_1>0 $$ and $$ R_2>0 $$, (38), (39), (41), and (43), one can show that the two sequences of $$ \left\lbrace L_{1}(x_{1, m})\right\rbrace _{m}^{+\infty} $$ and $$ \left\lbrace L_{2}(x_{1, m})\right\rbrace _{m}^{+\infty} $$ both monotonically decrease over the interval $$ \left(-\infty, x_{1}^{e1}\right) \cup (x_{1}^{e2}, +\infty) $$. Then, the switching maximum energy sequence of $$ \left\lbrace H(x_{m})\right\rbrace _{m}^{+\infty} $$ also decreases monotonically over the following time interval: $$ \left(-\infty, \, x_{1}^{e1}\right) $$.

On the other hand, one obtains from Lemma 2 and (72) that

Then, as $$ m\rightarrow \infty $$ and all the horizontal ordinates $$ x_{1, m} $$ of switching states $$ x_m $$ are contained in the interval $$ (-\infty, p_{\min}) \subset (-\infty, \, x_{1}^{e1}) $$, where $$ p_{\min} $$ and $$ x_{1}^{e1} $$ satisfy $$ p_{\min} {\leqslant} x_{1}^{e1}{\leqslant} v_{1}{\leqslant} x_{1}^{e2} $$. And it follows from (73) that the following holds true.

where $$ { } p_{\min} $$ and $$ g_2 $$ are the same as in (58) and (7), respectively.

By the squeeze theorem, one obtains from (73) and (74) that

where $$ { } p_{\min} $$ and $$ g_2 $$ are the same as in (58) and (7), respectively.

As $$ m\rightarrow \infty $$ and all the horizontal ordinates $$ x_{1, m} $$ of switching states $$ x_m $$ are contained in the interval as follows. $$ (p_{\max}, \, +\infty) \subset(x_{1}^{e2}, +\infty) $$, where $$ p_{\min} {\leqslant} x_{1}^{e1}{\leqslant} v_{1}{\leqslant} x_{1}^{e2}\leq p_{\max} $$. One knows that

It is obvious from (72) and (76) and the squeeze theorem that

where $$ { } p_{\max} $$ and $$ g_1 $$ are the same as in (58) and (7), respectively.

Since $$ L_1(p_{\max}) $$ and $$ L_2(p_{\min}) $$ are contained in the region $$ \Omega $$, it can be seen from (24), (73), (74), and (76) that

(S2) As $$ v_{2}\in\left(x_{1}^{e2}, +\infty\right) $$ and the trajectory $$ x(t)\in\mathbb{D}-\Omega $$, for any $$ t\in[t_{m}, t_{m+1}) $$, $$ m\in\mathbb{N} $$, the switching maximum energy sequence $$ H(x_m) $$ corresponding to switching states $$ x_m=[x_{1, m}\; x_{2, m}]^T\in\mathbb{R}^2 $$ and $$ m\in\mathbb{N} $$ is as follows.

where the two functions $$ L_1(\cdot) $$ and $$ L_2(\cdot) $$ are defined in (6). $$ g_1 $$ and $$ g_2 $$ are in (7).

Similar to that proof of Situation (S1), one obtains from Lemma 2 and the condition of $$ R_1>0 $$ and $$ R_2>0 $$, (6), (38), (39), (73), and (43) that (78) also holds for this Situation. It thus follows from the above considerations of Situations (S1) and (S2) that Statement $$ (ii) $$ holds for Case (a).

As for Case (b), it is easy to see from the conditions of Statement $$ (ii) $$ that system (1) under the switching line $$ l_1 $$ in (2) is region stable with respect to the region $$ \Omega $$ defined in (6). Meanwhile, (28) of Definition 3 follows from the fact that the system trajectory $$ x(t) $$ is always contained in the region $$ \Omega $$. Thus, all the conditions of Definition (3) are satisfied for system (1). Therefore, the correctness of Statement $$ (ii) $$ follows from Definition (3) for Case (b).

Finally, we show that Statement $$ (ii) $$ also holds for Case (c) as follows. Similar to that proof of the two cases (C1) and (C2) of Situation $$ (i) $$, one can show from (69) that Statement $$ (ii) $$ holds true for the case that the whole trajectory $$ x(t\colon t_{m_{l}}, x_{m_{l}}, i_{m_{l}}) $$ of system (1) is contained in $$ \mathbb{D}-\operatorname{int}(\Omega_{\varepsilon}) $$. On the other hand, it is obvious that as $$ \varepsilon\rightarrow0 $$, The state trajectory $$ x(t) $$ contained in the region $$ \Omega_{\varepsilon}-\Omega $$ will gradually converge into the region $$ \Omega $$.

Based on the above, by Definition 3, we obtain from the condition of $$ R_1>0 $$ and $$ R_2>0 $$, (32), (74), (76), and (78) that system (1) under the switching line $$ l_1 $$ in (2) is asymptotically region stable with respect to the region $$ \Omega $$. That is, Statement $$ (ii) $$ holds.

(2) We show the conclusion of Theorem 1 also holds for the case that $$ g_1= g_2 $$.

In this case, letting $$ L_{1}(x_{1})=L_{2}(x_{1}) $$, together with (6), yields

Solving the above equation (80), we obtain its solution denoted by $$ v_1 $$ as follows.

It can be obtained from (81) that $$ p_{\max}=2x_{1}^{e2}-x_{1}^{e1} $$ and $$ p_{\min}=2x_{1}^{e1}-x_{1}^{e2} $$ being the two points on the boundary of the region $$ \Omega $$. Then, similar to that of the case of $$ g_{1}\neq g_{2} $$, it can be shown that Theorem 1 also holds for the case that $$ g_1= g_2 $$.

Thus, the proof of Theorem 1 is finished. □

For the case that horizontal ordinates of the equilibrium points $$ x^{e1} $$ and $$ x^{e2} $$ of the two subsystems are the same but their vertical ordinates are different, i.e., $$ x_{1}^{e1}= x_{1}^{e2} $$ and $$ x_{2}^{e1}\neq x_{2}^{e2} $$, we present another main result of this paper as follows.

Theorem 2Consider system (1) with the compact region $$ \Psi $$ in (25) and the switching line $$ l_1 $$ in (2). For the case that $$ x_{2}^{e1}\neq x_{2}^{e2} $$, system (1) under the switching line $$ l_1 $$ in (2) is

(i) region stable with respect to the region $$ \Psi $$ in (25), if $$ J_1\not= 0 $$, $$ J_2\not=0 $$, $$ R_1{\geqslant} 0 $$, $$ R_2{\geqslant} 0 $$, and

where $$ \alpha=\min\big\{\lambda_{\min}(Q_1), \lambda_{\min}(Q_2)\big\} $$, $$ \beta=\max\big\{\lambda_{\max}(Q_1), \lambda_{\max}(Q_2)\big\} $$, and $$ v_{2} $$ is the intersection point of the two parabolic curves $$ M_{1}(x_{2}) $$ and $$ M_{2}(x_{2}) $$ in (8) over the interval $$ (x_{2}^{e1}, \, x_{2}^{e2}) $$ in the vertical axis, i.e.,

where $$ \eth _1 $$ and $$ \eth _2 $$ are defined in (9).

(ii) asymptotic region stable with respect to the region $$ \Omega $$, if $$ J_1\not= 0 $$, $$ J_2\not=0 $$, $$ R_1> 0 $$, $$ R_2> 0 $$, and the condition (82) are all satisfied.

Proof: Similar to that proof of Theorem 1, we show from $$ x_2^{e1}\neq x_2^{e2} $$ that Theorem 2 also holds via replacing all the horizontal ordinates $$ x_1^{e1} $$ and $$ x_1^{e2} $$ of the equilibrium points $$ x^{e1} $$ and $$ x^{e2} $$ of the two subsystems by their vertical ordinates of $$ x_2^{e1} $$ and $$ x_2^{e2} $$ during the whole proof of Theorem 1. □

4.1. An application to a switching DC electric circuit

Consider an ideal switching DC electric circuit, as shown in Figure 2. In which $$ L $$, $$ C $$, $$ r_{1} $$, $$ r_{2} $$, $$ E $$, and $$ VF $$ denote the inductor, the capacitor, the two resistors, the DC electric source, and the switch, respectively. For simplicity, the notations of $$ L $$, $$ C $$, $$ r_{1} $$, $$ r_{2} $$, and $$ E $$ also denote the inductance, the capacitance, the resistances, and the electric voltage, respectively, of the corresponding circuit elements. The constant parameters of these circuit elements are chosen as follows. $$ L=10mH $$, $$ C=200mF $$, $$ r_{1}=20\Omega $$, $$ r_{2}=60\Omega $$, and $$ E=3V $$. For the switch $$ VF $$, there are always the following two actions: the OFF and the ON. Therefore, such an electric circuit is essential to a switching DC electric circuit. By the famous Kirchhoff's voltage law and Kirchhoff's current law, we can model the electric circuit by a switched linear Hamiltonian system, such as system (1).

Figure 2. Switching circuit schematic.

To do that, we let $$ x=[x_1, \, x_2]^T $$ be the state of the switching DC electric circuit situated in Figure 2. Where the first element $$ x_1 $$ of the state $$ x $$ denotes the electric current $$ i_L(t) $$ of the inductor $$ L $$, and the second element $$ x_2 $$ of the state $$ x $$ denotes the voltage $$ U_C(t) $$ of the capacitor $$ C $$. Then, the electric circuit is modeled as a switched dissipative linear Hamiltonian system with two subsystems as follows. The following two subsystems are expressed as

where the corresponding matrices of the first and second subsystems are as follows.

The Hamiltonian functions of the two subsystems are, respectively, listed as follows.

in which the matrices $$ Q_1 $$ and $$ Q_2 $$ are listed in the following:

The equilibrium points of the first subsystem and the second subsystem are $$ x^{e1}=[0.05\; 3] ^T $$ and $$ x^{e2}=[0.2\; 3] ^T $$, respectively. The equilibrium points $$ x^{e1} $$ and $$ x^{e2} $$ of the two different subsystems are simultaneously passed through a switching line $$ l_1 $$ expressed as

which is a special switching path $$ \sigma(t) $$ governing the activation of the two subsystems in system (84).

One obtains from (85) and (87) that

and

From (89) and (90), it can be seen that although the two subsystems of system (84) are two essential linear systems, their state matrices $$ (J_1-R_1)Q_1 $$ and $$ (J_2-R_2)Q_2 $$ do not commute each other. Therefore, the region stability of system (84) cannot be verified by the stability criteria given in the reference ^[17]. Moreover, there are not any other stability criteria reported in the open literature that can be used to check the stability of system (84). However, by the main results of this paper, we can check the stability of system (84) as follows.

It can be obtained from (87), (7), (42), and the equilibrium points $$ x^{e1}=[0.05\; 3] ^T $$ and $$ x^{e2}=[0.2\; 3] ^T $$ of the two subsystems that

and

It is obvious from (91) and (92) that (41) of Theorem 1 is satisfied. One knows from (85) that the following hold: $$ J_{1}\not=0 $$, $$ J_{2}\not=0 $$, $$ R_1{\geqslant}0 $$, and $$ R_2{\geqslant}0 $$, which are also satisfied for system (84) under the switching line $$ l_1 $$ in (88). Therefore, all the conditions of Conclusion $$ (i) $$ of Theorem 1 are satisfied for system (84). By Theorem 1, we know that system (84) under the switching line $$ l_1 $$ in (88) is region stable with respect to the following region expressed as

To show the above conclusion, we will simulate system (84) as follows. An initial state is chosen as $$ x_{0}=[0.03\, \, 2]^T $$, which is contained in the exterior of the region $$ \Omega $$ in (93). The simulations are carried out and illustrated in Figures 3-5. Figure 3 denotes the response of switching path $$ \sigma(t) $$ in relation to the switching line $$ l_1 $$ in (88) with respect to time $$ t $$. Figure 4 denotes the trajectory of system (84) under the switching path $$ \sigma $$ in relation to the switching line $$ l_{1} $$ starting from the initial state $$ x_{0} $$ with respect to time $$ t $$. Figure 5 denotes the trajectory of system (84) under the switching path $$ \sigma $$ in relation to the switching line $$ l_{1} $$ in the plane $$ \mathbb{R}^{2} $$ starting from the initial state $$ x_{0} $$. It is easy to see from Figure 5 that the trajectory $$ x(t) $$ of system (84) goes to or into the region $$ \Omega $$ as $$ t\rightarrow +\infty $$. Therefore, the simulations show the effectiveness and practicality of Conclusion $$ (i) $$ of Theorem 1.

Figure 3. The response of the switching path $$ \sigma $$ in relation to the switching line $$ l_{1} $$ with respect to time $$ t $$.

Figure 4. The trajectory of system (84) under the switching path $$ \sigma $$ in relation to the switching line $$ l_{1} $$ starting from the initial state $$ x_{0} $$ with respect to time $$ t $$.

Figure 5. The trajectory of system (84) under the switching path $$ \sigma $$ in relation to the switching line $$ l_{1} $$ in the plane $$ \mathbb{R}^{2} $$ starting from the initial state $$ x_{0} $$.

4.2. Two numerical examples

Example 1Consider a switched linear Hamiltonian system with two subsystems as follows.

governed by a switching path $$ \sigma(t)\, :\, [0, \, +\infty)\rightarrow \{1, \, 2\} $$, which is also a switching line passing through the two points, $$ x^{e1}=[1\; 2] ^T $$ and $$ x^{e2}=[8\; 16] ^T $$, of the equilibrium points of the first subsystem and the second subsystem, respectively. That is, the switching line can be expressed as

In system (94), the Hamiltonian functions of the two subsystems are listed as

where the matrices $$ Q_1 $$ and $$ Q_2 $$ are, respectively, listed as the corresponding matrices of the subsystems as follows.

The corresponding matrices of the first and second subsystems are as follows.

We obtain from (97) and (98) that

and

From (99) and (100), it is obvious that although the two subsystems of system (94) are two essential linear systems, their state matrices $$ (J_1-R_1)Q_1 $$ and $$ (J_2-R_2)Q_2 $$ cannot commute. Therefore, the region stability of system (94) cannot be verified by the stability criteria obtained in the reference ^[17]. Moreover, there is not any stability criteria reported in the open literature. However, we can check the stability of system (94) as follows.

According to (7), (97), and (41), we obtain that

and

It is obvious from (101) and (102) that (41) of Theorem 1 is satisfied.

It is obvious from (98), (101), and (102) that $$ J_{1}\not=0 $$, $$ J_{2}\not=0 $$, $$ R_1>0 $$, $$ R_2>0 $$, and (41) are all satisfied for system (94) under the switching line $$ l_1 $$ denoted in (95). That is, all the conditions of Conclusion $$ (ii) $$ of Theorem 1 are satisfied for system (94). By Theorem 1, we know that system (94) under the switching line $$ l_1 $$ in (95) is asymptotically region stable with respect to the following region:

To show the above conclusion by simulations, we choose the following two initial states: $$ x_{01}=[-25\, \, 50]^T $$ and $$ x_{02}=[4\, \, 8]^{T} $$. It is easy to see that the two initial states $$ x_{01} $$ and $$ x_{02} $$ are contained in the interior and the exterior of the region $$ \Omega $$ in (103), respectively. The numerical simulations are then carried out, and the results are shown in Figures 6-11, which denote the trajectories of system (94) under the switching line $$ l_1 $$ in (95) starting from the two initial states in the plane $$ \mathbb{R}^2 $$, the trajectories with respect to time $$ t $$, and switching path with respect to time $$ t $$, respectively. It can be seen from Figures 6 and 7 that the trajectory $$ x(t) $$ of system (94) goes to/into the region $$ \Omega $$ as $$ t\rightarrow +\infty $$. Therefore, these simulations show that the Conclusion $$ (ii) $$ of Theorem 1 is effective and practical.

Figure 6. The trajectory of system (94) under the switching path $$ \sigma_1 $$ in relation to the switching line $$ l_1 $$ in the plane $$ \mathbb{R}^2 $$ starting from the initial state $$ x_{01} $$.

Figure 7. The trajectory of system (94) under the switching path $$ \sigma_2 $$ in relation to the switching line $$ l_1 $$ in the plane $$ \mathbb{R}^2 $$ starting from the initial state $$ x_{02} $$.

Figure 8. The trajectory of system (94) under the switching path $$ \sigma_1 $$ in relation to the switching line $$ l_1 $$ starting from the initial state $$ x_{01} $$ with respect to time $$ t $$.

Figure 9. The trajectory of system (94) under the switching path $$ \sigma_2 $$ in relation to the switching line $$ l_1 $$ starting from the initial state $$ x_{02} $$ with respect to time $$ t $$.

Figure 10. The response of the switching path $$ \sigma_1 $$ in relation to the switching line $$ l_1 $$ with respect to time $$ t $$.

Figure 11. The response of the switching path $$ \sigma_2 $$ in relation to the switching line $$ l_1 $$ with respect to time $$ t $$.

Example 2Consider a switched linear Hamiltonian system with two subsystems as follows.

governed by a switching path $$ \sigma(t)\, :\, [0, \, +\infty)\rightarrow \{1, \, 2\} $$, which is also a switching line passing through the two points, $$ x^{e1}=[2\; 2] ^T $$ and $$ x^{e2}=[2\; 12] ^T $$, of the equilibrium points of the first subsystem and the second subsystem, respectively. That is, the switching line can be expressed as

In system (104), the Hamiltonian functions of the two subsystems are listed as

where the matrices $$ Q_1 $$ and $$ Q_2 $$ are, respectively, listed as the corresponding matrices of the subsystems as follows.

The corresponding matrices of the first and second subsystems are as follows.

One obtains from (104), (107), and (108) that the following two:

and

From (109) and (110), it is obvious that although the two subsystems of system (104) are also two linear systems, their state matrices $$ (J_1-R_1)Q_1 $$ and $$ (J_2-R_2)Q_2 $$ are not commutative. Therefore, the region stability of system (104) cannot be verified by the stability criteria obtained in the reference ^[17]. Moreover, there is not any stability criteria reported in the open literature. However, we can check the stability of system (104) as follows.

According to (7), (107), and (41), we obtain that

and

It is obvious from (108), (111), and (112) that $$ J_{1}\not=0 $$, $$ J_{2}\not=0 $$, $$ R_1>0 $$, $$ R_2>0 $$, and (41) are all satisfied for system (104) under the switching line $$ l_1 $$ denoted in (105). That is, all the conditions of Conclusion $$ (ii) $$ of Theorem 2 are satisfied for system (104). By Theorem 2, we know that system (104) under the switching line $$ l_1 $$ in (105) is asymptotically region stable with respect to the following region:

To show the above conclusion by simulations, we choose the following two initial states: $$ x_{01}=[-20\, \, 50]^T $$ and $$ x_{02}=[2\, \, 20]^{T} $$. It is easy to see that the two initial states $$ x_{01} $$ and $$ x_{02} $$ are contained in the interior and the exterior of the region $$ \Psi $$ in (113), respectively. The numerical simulations are then carried out, and the results are shown in Figures 12-17, which denote the trajectories of system (104) under the switching line $$ l_1 $$ in (105) starting from the two initial states in the plane $$ \mathbb{R}^2 $$, the trajectories with respect to time $$ t $$, and switching path with respect to time $$ t $$, respectively. It can be seen from Figures 12 and 13 that the trajectories $$ x(t) $$ of system (104) goes to/into the region $$ \Omega $$ as $$ t\rightarrow +\infty $$. Therefore, these simulations show that the Conclusion $$ (ii) $$ of Theorem 2 is effective and practical.

Figure 12. The trajectory of system (104) under the switching path $$ \sigma_3 $$ in relation to the switching line $$ l_1 $$ in the plane $$ \mathbb{R}^2 $$ starting from the initial state $$ x_{01} $$.

Figure 13. The trajectory of system (104) under the switching path $$ \sigma_4 $$ in relation to the switching line $$ l_1 $$ in the plane $$ \mathbb{R}^2 $$ starting from the initial state $$ x_{02} $$.

Figure 14. The trajectory of system (104) under the switching path $$ \sigma_3 $$ in relation to the switching line $$ l_1 $$ starting from the initial state $$ x_{01} $$ with respect to time $$ t $$.

Figure 15. The trajectory of system (104) under the switching path $$ \sigma_4 $$ in relation to the switching line $$ l_1 $$ starting from the initial state $$ x_{02} $$ with respect to time $$ t $$.

Figure 16. The response of the switching path $$ \sigma_3 $$ in relation to the switching line $$ l_1 $$ with respect to time $$ t $$.

Figure 17. The response of the switching path $$ \sigma_4 $$ in relation to the switching line $$ l_1 $$ with respect to time $$ t $$.

Authors' contributions

Made substantial contributions to supervision, writing, review, editing, and methodology: Zhu L

Performed writing-original draft, software, validation, and visualization: Liu T

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Financial support and sponsorship

This work was supported by Shandong Natural Science Foundation of China under Grant (No. ZR2021MF012) and Cultivating Foundation of Qilu University of Technology under Grant (No. 2022PYI010).

Conflicts of interest

All authors declared that there are no conflicts of interest.

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© The Author(s) 2023.