CN112995997A - Charging wireless sensor network malicious program variation model and optimal control method thereof - Google Patents

Charging wireless sensor network malicious program variation model and optimal control method thereof Download PDF

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CN112995997A
CN112995997A CN202110171185.2A CN202110171185A CN112995997A CN 112995997 A CN112995997 A CN 112995997A CN 202110171185 A CN202110171185 A CN 202110171185A CN 112995997 A CN112995997 A CN 112995997A
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CN112995997B (en
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刘贵云
彭智敏
李君强
舒聪
梁倩韵
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Guangzhou University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/14—Network analysis or design
    • H04L41/145—Network analysis or design involving simulating, designing, planning or modelling of a network
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L63/00—Network architectures or network communication protocols for network security
    • H04L63/14—Network architectures or network communication protocols for network security for detecting or protecting against malicious traffic
    • H04L63/1441—Countermeasures against malicious traffic
    • H04L63/145—Countermeasures against malicious traffic the attack involving the propagation of malware through the network, e.g. viruses, trojans or worms
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W12/00—Security arrangements; Authentication; Protecting privacy or anonymity
    • H04W12/009—Security arrangements; Authentication; Protecting privacy or anonymity specially adapted for networks, e.g. wireless sensor networks, ad-hoc networks, RFID networks or cloud networks
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W84/00—Network topologies
    • H04W84/18—Self-organising networks, e.g. ad-hoc networks or sensor networks
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    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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Abstract

The invention discloses a charging wireless sensor network malicious program variation model and an optimal control method thereof, wherein the method comprises the steps of firstly constructing a network node state transition diagram of a charging wireless sensor network, wherein network nodes in the state transition diagram comprise susceptible nodes, infected nodes, variation nodes, low-energy nodes and dead nodes; then listing differential equation expressions of the nodes according to the state transition diagram of the network nodes; then, establishing a cost target function of optimal control according to a differential equation expression; then, establishing a Hamiltonian according to the cost objective function and the constraint condition; and finally, solving an optimal control result by utilizing a maximum value principle according to a Hamiltonian. According to the method, the node energy state and the malicious program infection and variation are considered, the malicious program variation condition in the real chargeable wireless sensor network can be accurately simulated, and the optimal control of the network is accurately and reasonably realized.

Description

Charging wireless sensor network malicious program variation model and optimal control method thereof
Technical Field
The invention relates to the technical field of rechargeable wireless sensor networks, in particular to a malicious program variation model of a rechargeable wireless sensor network and an optimal control method thereof.
Background
With the rapid development of computer networks, computer networks are being used in various aspects, making them very important in various fields. Similarly, the rechargeable wireless sensor network is also applied to a plurality of fields, and is favored by more and more people due to its mobility and rechargeable characteristic and flexible configuration. Therefore, the safety performance of the rechargeable wireless sensing network needs to be guaranteed to a certain extent, and only if the safety performance of the rechargeable wireless sensing network reaches a certain extent, more efficient and rapid service can be provided, so that the greatest benefit can be brought to the economic aspect.
Regarding the security problem of the rechargeable wireless sensor network, the biggest threat is the damage of a malicious program to the sensor nodes, and the malicious program can embed codes into the sensor nodes without being detected, so that the security and the integrity of infected data can be damaged during operation. In addition, a malicious program may have a "mutation", which is equivalent to giving an artificial life to a sensor, and like a mutation problem of a computer virus, there are unexpected mutations due to a copy problem, and a shell-replacement mutation method in which decryption is performed by a different decryptor due to encryption. Therefore, in order to guarantee the use safety of the rechargeable wireless sensor network, construct an accurate and appropriate malicious program variation model, and establish a corresponding optimal control strategy, it is necessary. The optimal control not only needs to be combined with the charging and discharging factors to carry out the optimal cost control, but also needs to consider the influence of the variant malicious programs on the rechargeable wireless sensor network.
Disclosure of Invention
The first objective of the present invention is to overcome the drawbacks and deficiencies of the prior art, and provide a malicious program variation model for a charging wireless sensor network, which can accurately simulate the malicious program variation in a real rechargeable wireless sensor network.
The second objective of the present invention is to provide an optimal control method for a rogue program mutation model of a rechargeable wireless sensor network, which can accurately and reasonably implement optimal control of the rechargeable wireless sensor network.
A third object of the present invention is to provide a computer-readable storage medium.
It is a fourth object of the invention to provide a computing device.
The first purpose of the invention is realized by the following technical scheme: a charging wireless sensor network malicious program variation model considers susceptible nodes S and infected nodes I in a charging wireless sensor network1Variant node I2Low energy node L and death node D, expressed as:
Figure BDA0002938974970000021
wherein t is time; l issThe nodes are vulnerable to malicious programs and are in a low-energy state;
Figure BDA0002938974970000022
the nodes which are successfully attacked by the malicious programs and are in a low-energy state;
Figure BDA0002938974970000023
the nodes which are successfully attacked by the variant malicious programs and are in a low-energy state;
c is the control proportion of the low energy state node returning to the normal energy state through charging; p is a control proportion of each state which consumes energy and converts the state into a low-energy state; d dead for each node stateA proportionality coefficient; beta is a1A propagation coefficient for infection of a susceptible node into an infected node; beta is a2The propagation coefficient of the susceptible node infected as a variable node; epsilon is a proportionality coefficient of an infected node mutated into a variant node; gamma ray1Searching and killing the control proportion of the malicious programs of the infected nodes; gamma ray2Searching and killing control proportion of the malicious programs of the variant nodes; d is the death proportion coefficient of each state node; and A is the supplementary number of new nodes in the rechargeable wireless sensor network.
The second purpose of the invention is realized by the following technical scheme: an optimal control method for a charging wireless sensor network malicious program variation model comprises the following steps:
s1, constructing a corresponding network node state transition diagram aiming at the chargeable wireless sensing network, wherein the network nodes comprise susceptible nodes S and infected nodes I1Variant node I2A low energy node L and a death node D;
s2, listing differential equation expressions of the nodes according to the state transition diagram of the network nodes;
s3, establishing a cost objective function of optimal control according to the differential equation expression;
s4, establishing a Hamiltonian according to the cost objective function and the constraint condition;
and S5, solving an optimal control result by utilizing a maximum value principle according to the Hamiltonian.
Preferably, the network node state transition graph is constructed based on the dynamic transmission of infectious diseases.
Preferably, the differential equation expression of the node is expressed as follows:
Figure BDA0002938974970000031
wherein t is time; l issThe nodes are vulnerable to malicious programs and are in a low-energy state;
Figure BDA0002938974970000032
is successfully attacked by a malicious programA node in a low energy state;
Figure BDA0002938974970000033
the nodes which are successfully attacked by the variant malicious programs and are in a low-energy state;
c is the control proportion of the low energy state node returning to the normal energy state through charging; p is a control proportion that each state node consumes energy and converts the node into a low-energy state; d is the death proportion coefficient of each state node; beta is a1A propagation coefficient for infection of a susceptible node into an infected node; beta is a2The propagation coefficient of infection of susceptible nodes into variable nodes; epsilon is a proportionality coefficient of an infected node mutated into a variant node; gamma ray1Searching and killing the control proportion of the malicious programs of the infected nodes; gamma ray2Searching and killing control proportion of the malicious programs of the variant nodes; d is the death proportion coefficient of each state node; and A is the supplementary number of new nodes in the rechargeable wireless sensor network.
Further, the process of step S3 for establishing the objective function according to the differential equation expression is as follows:
first, define the cost of charging the low-energy infected node as
Figure BDA0002938974970000041
The cost of charging the low-energy variant node is
Figure BDA0002938974970000042
The charging cost price of the low-energy susceptible node is QLSAt the cost of malicious program clean-up
Figure BDA0002938974970000043
The cost of cleaning up the variant malicious program is
Figure BDA0002938974970000044
And the charging ratio of the low-energy node is more than or equal to 0 and less than or equal to K1Less than or equal to 1, and the malicious program cleaning proportion is more than or equal to 0 and less than or equal to K2Less than or equal to 1, and the variable malicious program cleaning ratio is more than or equal to 0 and less than or equal to K3Less than or equal to 1; wherein the content of the first and second substances,
Figure BDA0002938974970000045
c＝K1；γ1＝K2；γ2＝K3；
then, designing a cost objective function minJ (u) meeting the control condition:
Figure BDA0002938974970000046
wherein the optimization control variable u is described as satisfying I1And I2And under the condition of the minimum number of the time tf, the node L with low susceptibilityS(tf) infecting nodes of low energy by malicious programs
Figure BDA0002938974970000047
Variant malicious program infecting low-energy nodes
Figure BDA0002938974970000048
There is a cost-optimal solution, i.e., the feasible region is [0,1 ] at time tf]Control variable set of
Figure BDA0002938974970000049
Program clear ratio.
Further, in step S4, the process of constructing the hamiltonian is as follows:
firstly, solving an extreme value problem of a cost target function by utilizing a Ponderland Riagingmaximum value principle:
Figure BDA00029389749700000410
wherein, x (t) is a state vector equation as an extreme value trajectory equation; alpha (t) is a co-modal vector equation comprising alpha1(t)～α6(t) as a constraint; alpha is alphaT(t) transposing the co-modal vector; h is a Hamiltonian expression, and the Hamiltonian actually is the problem of solving the minimum value under the condition that the target function is added into the constraint condition;
and c is K1；γ1＝K2；γ2＝K3Substituting it into the hamiltonian:
Figure BDA0002938974970000051
further, the process of step S5 is as follows:
solving optimal control by using maximum principle, the following covariate variable equation system { alpha ] is required to be satisfiedi(t) (i ═ 1,2,3,4,5,6) no variation }, yielding:
Figure BDA0002938974970000052
Figure BDA0002938974970000053
Figure BDA0002938974970000054
Figure BDA0002938974970000055
Figure BDA0002938974970000056
Figure BDA0002938974970000057
target I optimized at time tf of discussion1And I2In the case of (1), the cross-section condition of the final state needs to be satisfied while the covariate equation system is satisfied:
α1(tf)＝0；α2(tf)＝1；α3(tf)＝1；α4(tf)＝α5(tf)＝α6(tf)＝0
under the condition of meeting the cross-section condition, the variation of the control variable set is solved by utilizing a maximum value principle, and the following steps are obtained:
Figure BDA0002938974970000061
Figure BDA0002938974970000062
Figure BDA0002938974970000063
the formula is solved as follows:
Figure BDA0002938974970000064
Figure BDA0002938974970000065
Figure BDA0002938974970000066
therefore, the optimal control is obtained as follows:
Figure BDA0002938974970000067
Figure BDA0002938974970000068
Figure BDA0002938974970000069
the third purpose of the invention is realized by the following technical scheme: a computer-readable storage medium stores a program, which when executed by a processor, implements the optimal control method for a charging wireless sensor network rogue program variation model according to the first object of the present invention.
The fourth purpose of the invention is realized by the following technical scheme: the computing device comprises a processor and a memory for storing processor executable programs, and when the processor executes the programs stored in the memory, the optimal control method for the charging wireless sensor network malicious program variation model is realized.
Compared with the prior art, the invention has the following advantages and effects:
(1) the charging wireless sensor network malicious program variation model constructed by the invention can combine the node energy state and the malicious program infection and variation, so that the method can be closer to reality and can more accurately simulate the malicious program variation condition in the real charging wireless sensor network.
(2) The method considers the states of various nodes such as susceptibility, infection, variation, low energy and death, and takes the minimum number of infected nodes and variation nodes as an optimization target to calculate the optimal solution of the cost of the low energy nodes infected by the susceptibility, the low energy nodes infected by the malicious programs and the varied malicious programs, so that the rechargeable wireless sensor network can carry out global regulation (charging and killing) according to the charging proportion, the malicious program removing proportion and the varied malicious program removing proportion of the low energy nodes under optimal control, the lowest cost is realized, meanwhile, the infection and variation of the malicious programs in the network are inhibited, and the availability and the safety of the network are improved.
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Fig. 1 is a flowchart of an optimal control method for a malicious program mutation model in a charging wireless sensor network according to the present invention.
Fig. 2 is a schematic diagram of a state transition diagram of a wireless sensor network.
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but the present invention is not limited thereto.
Example 1
The embodiment discloses an optimal control method for a charging wireless sensor network malicious program variation model, as shown in fig. 1, the optimal control method comprises the following steps:
and S1, constructing a corresponding network node state transition diagram for the chargeable wireless sensing network. Here, the network node state transition diagram is constructed according to the transmission of the dynamics of infectious diseases, and the rechargeable wireless sensing network comprises a susceptible node S and an infected node I1Variant node I2Low-energy node L and dead node D, therefore, the network node state transition diagram correspondingly contains susceptible node S and infected node I1Variant node I2Low energy node L and dead node D, see fig. 2.
S2, listing differential equation expressions of the nodes according to the state transition diagram of the network nodes, and expressing the differential equation expressions as follows:
Figure BDA0002938974970000081
the differential equation expression is a malicious program variation model of the rechargeable wireless sensor network, wherein t is time; l issNodes that are vulnerable to malicious programs and are in a low energy state (which may be referred to as vulnerable low energy nodes);
Figure BDA0002938974970000082
the node which is successfully attacked by the malicious program and is in a low-energy state (the node which is infected by the malicious program can be called as a node with low energy infection);
Figure BDA0002938974970000083
nodes which are successfully attacked by the variant malware and are in a low-energy state (which can be called variant malware infected low-energy nodes);
c is the control proportion of the low energy state node returning to the normal energy state through charging; p is a control proportion that each state node consumes energy and converts the node into a low-energy state; d is the death proportion coefficient of each state node;β1a propagation coefficient for infection of a susceptible node into an infected node; beta is a2The propagation coefficient of infection of susceptible nodes into variable nodes; epsilon is a proportionality coefficient of an infected node mutated into a variant node; gamma ray1Searching and killing the control proportion of the malicious programs of the infected nodes; gamma ray2Searching and killing control proportion of the malicious programs of the variant nodes; d is the death proportion coefficient of each state node; and A is the supplementary number of new nodes in the rechargeable wireless sensor network.
S3, establishing a cost objective function of optimal control according to a differential equation expression:
first, define the cost of charging the low-energy infected node as
Figure BDA0002938974970000084
The cost of charging the low-energy variant node is
Figure BDA0002938974970000085
The charging cost price of the low-energy susceptible node is QLSAt the cost of malicious program clean-up
Figure BDA0002938974970000086
The cost of cleaning up the variant malicious program is
Figure BDA0002938974970000087
And the charging ratio of the low-energy node is more than or equal to 0 and less than or equal to K1Less than or equal to 1, and the malicious program cleaning proportion is more than or equal to 0 and less than or equal to K2Less than or equal to 1, and the variable malicious program cleaning ratio is more than or equal to 0 and less than or equal to K3Less than or equal to 1; wherein the content of the first and second substances,
Figure BDA0002938974970000088
c＝K1；γ1＝K2；γ2＝K3；
then, designing a cost objective function minJ (u) meeting the control condition:
Figure BDA0002938974970000089
wherein the optimization control variable u is described as satisfying I1And I2And under the condition of the minimum number of the time tf, the node L with low susceptibilityS(tf) infecting nodes of low energy by malicious programs
Figure BDA0002938974970000091
Variant malicious program infecting low-energy nodes
Figure BDA0002938974970000092
There is a cost-optimal solution, i.e., the feasible region is [0,1 ] at time tf]Control variable set of
Figure BDA0002938974970000093
Program clear ratio.
S4, establishing a Hamiltonian according to the cost objective function and the constraint condition:
firstly, solving an extreme value problem of a cost target function by utilizing a Ponderland Riagingmaximum value principle:
Figure BDA0002938974970000094
wherein, x (t) is a state vector equation as an extreme value trajectory equation; alpha (t) is a co-modal vector equation comprising alpha1(t)～α6(t) as a constraint; alpha is alphaT(t) transposing the co-modal vector; h is Hamiltonian expression and is divided into an objective part and a constraint part, wherein G is the expression of the objective function, and alpha isT(t) f is a table of constraints, f representing the constraints in the form of differential equations;
and c is K1；γ1＝K2；γ2＝K3Substituting it into the hamiltonian:
Figure BDA0002938974970000095
the Hamiltonian is a problem of solving a minimum value when an objective function is added to a constraint condition, and the objective function is equivalent to an unconstrained function after being substituted into the constraint condition.
S5, solving an optimal control result by utilizing a maximum value principle according to a Hamiltonian:
first, the maximum principle is used to solve the optimal control, and the following set of covariate equations { alpha ] needs to be satisfiedi(t) (i ═ 1,2,3,4,5,6) no variation }, yielding:
Figure BDA0002938974970000101
Figure BDA0002938974970000102
Figure BDA0002938974970000103
Figure BDA0002938974970000104
Figure BDA0002938974970000105
Figure BDA0002938974970000106
target I optimized at time tf of discussion1And I2In the case of (1), not only needs to be satisfied by the covariate equation set, but also needs to satisfy the cross-section condition of the final state:
α1(tf)＝0；α2(tf)＝1；α3(tf)＝1；α4(tf)＝α5(tf)＝α6(tf)＝0
under the condition of meeting the cross-section condition, the variation of the control variable set is solved by utilizing a maximum value principle, and the following steps are obtained:
Figure BDA0002938974970000107
Figure BDA0002938974970000108
Figure BDA0002938974970000109
the formula is solved as follows:
Figure BDA0002938974970000111
Figure BDA0002938974970000112
Figure BDA0002938974970000113
therefore, the optimal control is obtained as follows:
Figure BDA0002938974970000114
Figure BDA0002938974970000115
Figure BDA0002938974970000116
example 2
The embodiment discloses a computer-readable storage medium, which stores a program, and when the program is executed by a processor, the method for optimally controlling a charging wireless sensor network malicious program variation model in embodiment 1 is implemented, specifically:
s1, constructing a corresponding network node state transition diagram aiming at the chargeable wireless sensing network, wherein the network nodes comprise susceptible nodes S and infected nodes I1Variant node I2A low energy node L and a death node D;
s2, listing differential equation expressions of the nodes according to the state transition diagram of the network nodes;
s3, establishing a cost objective function of optimal control according to the differential equation expression;
s4, establishing a Hamiltonian according to the cost objective function and the constraint condition;
and S5, solving an optimal control result by utilizing a maximum value principle according to the Hamiltonian.
The computer-readable storage medium in this embodiment may be a magnetic disk, an optical disk, a computer Memory, a Read-Only Memory (ROM), a Random Access Memory (RAM), a usb disk, a removable hard disk, or other media.
Example 3
The embodiment discloses a computing device, which includes a processor and a memory for storing an executable program of the processor, and when the processor executes the program stored in the memory, the optimal control method for the charging wireless sensor network malicious program variation model in embodiment 1 is implemented, specifically:
s1, constructing a corresponding network node state transition diagram aiming at the chargeable wireless sensing network, wherein the network nodes comprise susceptible nodes S and infected nodes I1Variant node I2A low energy node L and a death node D;
s2, listing differential equation expressions of the nodes according to the state transition diagram of the network nodes;
s3, establishing a cost objective function of optimal control according to the differential equation expression;
s4, establishing a Hamiltonian according to the cost objective function and the constraint condition;
and S5, solving an optimal control result by utilizing a maximum value principle according to the Hamiltonian.
The computing device described in this embodiment may be a desktop computer, a notebook computer, a smart phone, a PDA handheld terminal, a tablet computer, or other terminal device with a processor function.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.

Claims (9)

1. The charging wireless sensor network malicious program variation model is characterized in that the variation model considers susceptible nodes S and infected nodes I in the charging wireless sensor network1Variant node I2Low energy node L and death node D, expressed as:
Figure FDA0002938974960000011
wherein t is time; l issThe nodes are vulnerable to malicious programs and are in a low-energy state;
Figure FDA0002938974960000012
the nodes which are successfully attacked by the malicious programs and are in a low-energy state;
Figure FDA0002938974960000013
the nodes which are successfully attacked by the variant malicious programs and are in a low-energy state;
c is the control proportion of the low energy state node returning to the normal energy state through charging; p is a control proportion of each state which consumes energy and converts the state into a low-energy state; d is the proportional coefficient of death of each node state; beta is a1A propagation coefficient for infection of a susceptible node into an infected node; beta is a2Sensing of susceptible node infection as variant nodeA broadcast coefficient; epsilon is a proportionality coefficient of an infected node mutated into a variant node; gamma ray1Searching and killing the control proportion of the malicious programs of the infected nodes; gamma ray2Searching and killing control proportion of the malicious programs of the variant nodes; d is the death proportion coefficient of each state node; and A is the supplementary number of new nodes in the rechargeable wireless sensor network.
2. An optimal control method for a charging wireless sensor network malicious program variation model is characterized by comprising the following steps:
s1, constructing a corresponding network node state transition diagram aiming at the chargeable wireless sensing network, wherein the network nodes comprise susceptible nodes S and infected nodes I1Variant node I2A low energy node L and a death node D;
s2, listing differential equation expressions of the nodes according to the state transition diagram of the network nodes;
s3, establishing a cost objective function of optimal control according to the differential equation expression;
s4, establishing a Hamiltonian according to the cost objective function and the constraint condition;
and S5, solving an optimal control result by utilizing a maximum value principle according to the Hamiltonian.
3. The optimal control method according to claim 2, characterized in that the network node state transition graph is constructed from the transmission of the dynamics of infectious diseases.
4. The optimum control method according to claim 2, wherein the differential equation expression of the node is expressed as follows:
Figure FDA0002938974960000021
wherein t is time; l issThe nodes are vulnerable to malicious programs and are in a low-energy state;
Figure FDA0002938974960000022
the nodes which are successfully attacked by the malicious programs and are in a low-energy state;
Figure FDA0002938974960000023
the nodes which are successfully attacked by the variant malicious programs and are in a low-energy state;
c is the control proportion of the low energy state node returning to the normal energy state through charging; p is a control proportion that each state node consumes energy and converts the node into a low-energy state; d is the death proportion coefficient of each state node; beta is a1A propagation coefficient for infection of a susceptible node into an infected node; beta is a2The propagation coefficient of infection of susceptible nodes into variable nodes; epsilon is a proportionality coefficient of an infected node mutated into a variant node; gamma ray1Searching and killing the control proportion of the malicious programs of the infected nodes; gamma ray2Searching and killing control proportion of the malicious programs of the variant nodes; d is the death proportion coefficient of each state node; and A is the supplementary number of new nodes in the rechargeable wireless sensor network.
5. The optimum control method according to claim 4, wherein the step S3 of establishing the objective function according to the differential equation expression is as follows:
first, define the cost of charging the low-energy infected node as
Figure FDA0002938974960000024
The cost of charging the low-energy variant node is
Figure FDA0002938974960000025
The charging cost price of the low-energy susceptible node is QLSAt the cost of malicious program clean-up
Figure FDA0002938974960000026
The cost of cleaning up the variant malicious program is
Figure FDA0002938974960000027
And the charging ratio of the low-energy node is more than or equal to 0 and less than or equal to K1Less than or equal to 1, and the malicious program cleaning proportion is more than or equal to 0 and less than or equal to K2Less than or equal to 1, and the variable malicious program cleaning ratio is more than or equal to 0 and less than or equal to K3Less than or equal to 1; wherein the content of the first and second substances,
Figure FDA0002938974960000031
c＝K1；γ1＝K2；γ2＝K3；
then, designing a cost objective function minJ (u) meeting the control condition:
Figure FDA0002938974960000032
wherein the optimization control variable u is described as satisfying I1And I2And under the condition of the minimum number of the time tf, the node L with low susceptibilityS(tf) infecting nodes of low energy by malicious programs
Figure FDA0002938974960000033
Variant malicious program infecting low-energy nodes
Figure FDA0002938974960000034
There is a cost-optimal solution, i.e., the feasible region is [0,1 ] at time tf]Control variable set of
Figure FDA0002938974960000035
Program clear ratio.
6. The optimum control method according to claim 4, wherein in step S4, the procedure for constructing the Hamiltonian is:
firstly, solving an extreme value problem of a cost target function by utilizing a Ponderland Riagingmaximum value principle:
Figure FDA0002938974960000036
wherein, x (t) is a state vector equation as an extreme value trajectory equation; alpha (t) is a co-modal vector equation comprising alpha1(t)～α6(t) as a constraint; alpha is alphaT(t) transposing the co-modal vector; h is a Hamiltonian expression, and the Hamiltonian actually is the problem of solving the minimum value under the condition that the target function is added into the constraint condition;
and c is K1；γ1＝K2；γ2＝K3Substituting it into the hamiltonian:
Figure FDA0002938974960000041
7. the optimum control method according to claim 4, wherein the process of step S5 is as follows:
solving optimal control by using maximum principle, the following covariate variable equation system { alpha ] is required to be satisfiedi(t) (i ═ 1,2,3,4,5,6) no variation }, yielding:
Figure FDA0002938974960000042
Figure FDA0002938974960000043
Figure FDA0002938974960000044
Figure FDA0002938974960000045
Figure FDA0002938974960000046
Figure FDA0002938974960000047
target I optimized at time tf of discussion1And I2In the case of (1), the cross-section condition of the final state needs to be satisfied while the covariate equation system is satisfied:
α1(tf)＝0；α2(tf)＝1；α3(tf)＝1；α4(tf)＝α5(tf)＝α6(tf)＝0
under the condition of meeting the cross-section condition, the variation of the control variable set is solved by utilizing a maximum value principle, and the following steps are obtained:
Figure FDA0002938974960000051
Figure FDA0002938974960000052
Figure FDA0002938974960000053
the formula is solved as follows:
Figure FDA0002938974960000054
Figure FDA0002938974960000055
Figure FDA0002938974960000056
therefore, the optimal control is obtained as follows:
Figure FDA0002938974960000057
Figure FDA0002938974960000058
Figure FDA0002938974960000059
8. a computer-readable storage medium storing a program, wherein the program, when executed by a processor, implements the optimal control method for a rogue program variation model in a charging wireless sensor network according to any one of claims 2 to 7.
9. A computing device comprising a processor and a memory for storing processor-executable programs, wherein the processor, when executing the programs stored in the memory, implements the optimal control method for the charging wireless sensor network malware variation model of any one of claims 2 to 7.
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