Abstract

In control loops, inputs are often subject to sensor limitations. For systems governed by partial differential equations (PDEs) evolving in both space and time, a major limitation is the spatial range of sensing, as it can be impractical to measure the state simultaneously at every spatial point. Another limitation is the energy cost associated with always-on sensors, as it can be unaffordable to run the sensor all the time. In this work, we propose an intermittent-sensing scenario that takes into account the aforementioned limitations, in which the system’s state is measured at certain spatial subdomains over specific time intervals, rather than continuously across the entire spatial domain. This scenario depicts situations of restricted-energy sensors [1, 2], network control systems [3, 4], and mobile sensors [5].

The proposed sensing scenario is considered in the context of the perturbed KS equation given by [6, 7]

where \( x \in (0,1)\) is the one-dimensional space variable, \( \lambda >0\) is known as the destabilizing coefficient, and \( f\) is the perturbation. This equation has been used to model various physical phenomena including wildfires [8], turbulence in reaction-diffusion systems [6], and plasma instabilities [9], among others.

Since [10], several boundary controllers are proposed to stabilize the origin for (1) when \( f\equiv 0\) ; see [11, 12, 13]. These works, however, either assume a small \( \lambda\) , in which case boundary measurements are enough to ensure global asymptotic/exponential stability, or assume arbitrary \( \lambda\) but small initial conditions while requiring full state measurement. Additionally, \( \lambda\) is often assumed constant, though it could be space-dependent according to [14, 15]. To the best of our knowledge, the first work to study (1) without constraining the range of the initial condition nor the size of \( \lambda\) is [16]. The state in the latter reference is assumed to be measured intermittently over the subdomains \( (0,Y)\) and \( (Y,1)\) . Furthermore, feedback controllers are applied at \( x=0\) and \( x=1\) , while enforcing a zero state at \( x=Y\) , to ensure \( L^2\) global exponential stability of the origin. However, the destabilizing coefficient \( \lambda\) is assumed constant and known. This requirement is relaxed in [17] by allowing \( \lambda\) to be unknown, and proposing an adaptive design of the control parameters, to guarantee the same stability conclusions. On the other hand, the perturbed KS equation is considered in [18], where a boundary controller using boundary measurements is shown to guarantee an input-to-state stability with respect to \( f\) , under small \( \lambda\) .

In this work, we study boundary stabilization for (1) under intermittent sensing, in the presence of the bounded space- and time-dependent perturbation \( f\) . More specifically, we extend the adaptive Lyapunov-based approach in [17] to the perturbed case, when an upper bound on the norm of \( f\) is either known or unknown. In the former case, we establish \( L^2\) -input-to-state stability [19] with respect to \( f\) , while in the latter case, we establish only \( L^2\) -global uniform ultimate boundedness. On the other hand, under full state measurement, we simplify our design to achieve convergence to arbitrarily-small neighborhoods of the origin, even if we ignore an upper bound on the norm of \( f\) . Notably, the presence of perturbations prevents us from establishing the same properties under full and intermittent sensing, different from the unperturbed case, where \( L^2\) global exponential stability of the origin is guaranteed under both scenarios.

The rest of the paper is organized as follows. Problem formulation is in Section 2. The proposed adaptive controller as well as key Lyapunov inequalities are in Section 3. The results are in Sections 4 and 5. Finally, numerical simulations are in Section 6.

Notation. Depending on the context, a.e. means either almost every or almost everywhere. We denote by \( L^2(a,b)\) , for \( b>a\) , the space of functions \( u:[a,b]\to \mathbb{R}\) such that \( \int_{a}^{b}u(x)^2dx<+\infty\) . Furthermore, we let \( |u|_{\infty} := \text{ess sup}_{x\in (a,b)}|u(x)|:=\inf \{M\geq :|u(x)|\leq M\ \text{for a.e.}x\in (a,b)\}\) . For \( (x,t) \mapsto u(x,t)\) , the partial derivative of \( u\) with respect to \( t\) is denoted by \( u_t\) , the first partial derivative with respect to \( x\) is denoted by \( u_x\) , the second partial derivative with respect to \( x\) is denoted by \( u_{xx}\) (and so on), and we may write \( u(x)\) instead of \( u(x,t)\) . We denote the time derivative of a function \( t \mapsto V(t)\) by \( \dot{V}\) . We also denote the space derivative of a function \( x \mapsto \lambda(x)\) of a scalar variable by \( \lambda'\) . A continuous function \( \Phi : \mathbb{R}_{\geq 0}\to \mathbb{R}_{\geq 0}\) is of class \( \mathcal{K}\) if it vanishes at zero and is strictly increasing. Finally, for \( x \in \mathbb{R}\) , \( \text{sgn} (x) = 1\) if \( x>0\) , \( =0\) if \( x=0\) and \( =-1\) if \( x<0\) .

In this section, we formulate the proposed sensing scenario and the in-domain and boundary conditions.

2.1 Intermittent sensing

2.2 Control locations

We follow in this work a Lyapunov-based approach to design \( (u_1, u_2)\) for (2). To do so, we start introducing the Lyapunov function candidates

The following lemma establishes key Lyapunov inequalities along (2), that hold under the following assumption.

In view of (5)-(6), the control input \( u_1\) affects the (worst-case) behavior of \( V_1\) only and since \( (V_1,w_{xxx}(0))\) is available on \( I_1\) only, we propose to design \( u_1\) to guarantee a stable behavior for \( V_1\) over \( I_1\) and set \( u_1 = 0\) otherwise. For the same reasons, we design \( u_2\) to guarantee as stable behavior for \( V_2\) over \( I_2\) and set \( u_2 = 0\) otherwise. That is, we let

where \( \hat{\theta}_1, \hat{\theta}_2 \geq 0\) are adaptation parameters that we design to handle the fact that \( \lambda\) , \( \lambda'\) , and \( (\overline{T}_{1},\overline{T}_{2},\underline{T}_{1}, \underline{T}_{2})\) from Assumption 1 can be unknown. Furthermore, we design \( \kappa : \mathbb{R}_{\geq 0}\times \mathbb{R}\times \mathbb{R}_{\geq 0}\to \mathbb{R}\) to verify

The latter inequality combined to (5)-(6) yields

As a result, the stability analysis of the origin for the system of PDEs (2) is achieved through the stability analysis of the origin for the switched system (9).

We provide in the next lemma an example of a feedback law \( \kappa\) verifying (8).

In this section, we design the parameters \( (\hat{\theta}_1,\hat{\theta}_2)\) in (7) when \( \bar{f}\) is unknown. Then, to ensure stronger properties, we assume that \( \bar{f}\) is known.

4.1 Unknown perturbation range

4.2 Known perturbation range

In this section, we consider (1) subject to

where \( u_1\) is a control input to be designed. We assume that \( \lambda\) is absolutely continuous on \( (0,1)\) and that \( u\) is measured a.e. on \( (0,1)\) and for all time. As a result, defining the Lyapunov function candidate

and using Lemma 1, we conclude that, along (1) and (28),

where \( \theta := |\lambda'|_{\infty}^{2}+2\left(|\lambda|_{\infty}+\frac{1}{2}\right)\left(\left(|\lambda|_{\infty}+\frac{1}{2}\right)+12\right).\)

Now, by letting \( u_1 := \kappa (V,u_{xxx}(0),\hat{\theta})\) , where \( \kappa\) is defined in (10) and \( \hat{\theta} > 0\) to be designed, we conclude using (30) and Lemma 2 that

Finally, using Young inequality, we obtain

At this point, we propose to update the parameter \( \hat{\theta}\) according to the following algorithm.

According to Algorithm 3, if \( t\mapsto \hat{\theta}(t)\) is bounded, being non-decreasing, we can show that \( \hat{\theta}\) becomes constant after some finite time. Hence, according to (IX) in Algorithm 3, the function \( t\mapsto V(t)\) converges to a ball of radius \( \epsilon /\sigma\) . Since such a radius can be made arbitrarily small, we conclude that we can achieve the convergence of \( V\) to an arbitrarily small neighborhood of the origin.

The system (1) under (3) is simulated with the controller introduced in Section 4.1. We set \( \lambda(x) := 4\pi^{2}/0.25+50+2\sin(4x)^2\) for all \( x \in (0,Y)\) and \( \lambda(x) := \lambda (x-Y)\) for all \( x\in (Y,1)\) and \( f(x,t) := 12\times 10^3\sin (2\times 10^4t)+\xi(x,t)\) , where \( \xi\) is a white noise of power \( 80\) dBW generated using the Matlab function ’wgn’. We set \( I_{1} := [0,1)\cup [2,2.8)\cup [3.9,5)\cup [5.5,6.5)\cup [7,7.6) \times 10^{-3}\) and \( I_{2} := [1,2)\cup [2.8,3.9)\cup [5,5.5)\cup [6.5,7)\cup [7.6,8) \times 10^{-3}\) . Finally, \( (u_1,u_2)\) are given by (7), with \( \kappa\) defined in (10) for \( (\varepsilon, \delta) := (1,2)\) , and \( \hat{\theta}_1\) and \( \hat{\theta}_2\) designed according to Algorithm 1 with \( \hat{\theta}_1(0)=\hat{\theta}_2(0)=0\) , \( \Delta_1=\Delta_2 =0.01\) , and \( \sigma = 100\) .

The system is discretized with the mesh-free collocation method in [25], based on radial basis functions (RBFs). We estimate the boundary values \( w_{xxx}(0)\) and \( w_{x}(0)\) using the Euler forward scheme and \( v_{xxx}(1)\) and \( v_{x}(1)\) using the Euler backward scheme. The Lyapunov function candidates \( (V_{1},V_{2})\) are estimated using Riemannian sums.

We use multiquadric RBFs, which depend on a shape parameter \( c :=0.4\) . The initial and final simulation times are selected as \( t_{1} := 0\) and \( t_{f} := 8 \times 10^{-3}\) , respectively. We select \( 10\) uniformly separated collocation points on the interval \( [0,Y]\) (with \( Y:=0.5\) ), ranging from \( x_{o} := 0\) to \( x_{9} := Y\) , and select the same number of points on the interval \( [Y,1]\) . Hence, our spatial step is \( \Delta x \approx 0.05\) , which makes us choose the time step of \( \Delta t := 10^{-7}\) to keep the ratio \( \Delta t/\Delta x^4\) small. For comparison, the time step \( \Delta t := 10^{-9}\) and the spatial step \( \Delta x := 5\times 10^{-3}\) are used in [26], to simulate the third-order Korteweg-de Vries-Burgers’ equation defined on \( [0,1]\) . Finally, the initial condition \( u_o(x) := -A(\cos (4\pi x)-1)\) for all \( x \in (0,1)\) is tested for different values of \( A>0\) .

Figure 1 shows the closed-loop solutions corresponding to the initial condition \( u_o\) under \( A = 3\) . The corresponding control inputs are shown in Figure 2. As expected, the inputs are bounded, at the price of some chattering due to the discontinuous nature of our controller. Furthermore, the corresponding plots of \( t \mapsto (V_1(t),V_2(t))\) and \( t \mapsto V_1(t)+V_2(t)\) are depicted in Figure 3. On every interval in \( I_2\) , where \( u_1 = 0\) , the final value of \( V_1\) is greater than its initial value at the beginning of that time interval. This increase is compensated over the next intervals in \( I_1\) , where \( u_1 = \kappa (V_1,w_{xxx}(0),\hat{\theta}_1)\) . The same behavior is observed for \( V_2\) , which increases over intervals in \( I_1\) and decreases over intervals in \( I_2\) . In Figure 4, we plot the evolution of the adaptation parameters \( t \mapsto (\hat{\theta}_{1}(t),\hat{\theta}_{2}(t))\) , which shows the existence of an adaptation phase, during which \( \hat{\theta}_1\) and \( \hat{\theta}_2\) increase before they become, as expected, constant. Finally, Figure 5 illustrate the evolution of \( t \mapsto V_{1}(t) + V_{2}(t)\) for the initial condition \( u_o\) with \( A \in \{2,3,4,5,6,7\}\) . The conclusions of Theorem 1 are in agreement with Figure 5, which shows finite-time convergence of \( V_{1}+V_{2}\) to a bound that is independent of the various choices of \( A\) .

In this paper, we studied the boundary stabilization problem for the perturbed KS equation under an intermittent sensing scenario, and compared the obtained results with those we obtained with full sensing. The intermittent sensing scenario forces us to consider in-domain conditions, that we do not necessarily require under full sensing. Our design is Lyapunov-based, tailored to some adaptive design to handle unknown parameters and perturbations. The obtained results are summarized in the following table.

Several challenging research directions emerge from our work. A primary extension would involve considering measurements taken intermittently over an arbitrary number of subregions, rather than just two. Another compelling direction concerns uncertainty in the location \( Y\) , which becomes particularly relevant when the Lyapunov functions \( (V_1,V_2)\) must be approximated numerically using sensor measurements. A related scenario arises when enforcing the in-domain condition at \( x=Y\) while gathering measurements over intervals \( [0,Y_o]\) and \( [Y_o,1]\) with \( Y_o<Y\) . The consideration of packet losses under an average dwell-time condition is also an interesting research perspective. Finally, extending our approach to accommodate measurements from scanning and pointwise sensors [21, 27] represents a natural progression of our work.

Proof of Lemma 1

Proof of Lemma 2